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<FONT color="green">001</FONT>    /*<a name="line.1"></a>
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<FONT color="green">012</FONT>     * distributed under the License is distributed on an "AS IS" BASIS,<a name="line.12"></a>
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<FONT color="green">017</FONT>    package org.apache.commons.math3.linear;<a name="line.17"></a>
<FONT color="green">018</FONT>    <a name="line.18"></a>
<FONT color="green">019</FONT>    import org.apache.commons.math3.exception.DimensionMismatchException;<a name="line.19"></a>
<FONT color="green">020</FONT>    import org.apache.commons.math3.exception.MaxCountExceededException;<a name="line.20"></a>
<FONT color="green">021</FONT>    import org.apache.commons.math3.exception.NullArgumentException;<a name="line.21"></a>
<FONT color="green">022</FONT>    import org.apache.commons.math3.exception.util.ExceptionContext;<a name="line.22"></a>
<FONT color="green">023</FONT>    import org.apache.commons.math3.util.FastMath;<a name="line.23"></a>
<FONT color="green">024</FONT>    import org.apache.commons.math3.util.IterationManager;<a name="line.24"></a>
<FONT color="green">025</FONT>    import org.apache.commons.math3.util.MathUtils;<a name="line.25"></a>
<FONT color="green">026</FONT>    <a name="line.26"></a>
<FONT color="green">027</FONT>    /**<a name="line.27"></a>
<FONT color="green">028</FONT>     * &lt;p&gt;<a name="line.28"></a>
<FONT color="green">029</FONT>     * Implementation of the SYMMLQ iterative linear solver proposed by &lt;a<a name="line.29"></a>
<FONT color="green">030</FONT>     * href="#PAIG1975"&gt;Paige and Saunders (1975)&lt;/a&gt;. This implementation is<a name="line.30"></a>
<FONT color="green">031</FONT>     * largely based on the FORTRAN code by Pr. Michael A. Saunders, available &lt;a<a name="line.31"></a>
<FONT color="green">032</FONT>     * href="http://www.stanford.edu/group/SOL/software/symmlq/f77/"&gt;here&lt;/a&gt;.<a name="line.32"></a>
<FONT color="green">033</FONT>     * &lt;/p&gt;<a name="line.33"></a>
<FONT color="green">034</FONT>     * &lt;p&gt;<a name="line.34"></a>
<FONT color="green">035</FONT>     * SYMMLQ is designed to solve the system of linear equations A &amp;middot; x = b<a name="line.35"></a>
<FONT color="green">036</FONT>     * where A is an n &amp;times; n self-adjoint linear operator (defined as a<a name="line.36"></a>
<FONT color="green">037</FONT>     * {@link RealLinearOperator}), and b is a given vector. The operator A is not<a name="line.37"></a>
<FONT color="green">038</FONT>     * required to be positive definite. If A is known to be definite, the method of<a name="line.38"></a>
<FONT color="green">039</FONT>     * conjugate gradients might be preferred, since it will require about the same<a name="line.39"></a>
<FONT color="green">040</FONT>     * number of iterations as SYMMLQ but slightly less work per iteration.<a name="line.40"></a>
<FONT color="green">041</FONT>     * &lt;/p&gt;<a name="line.41"></a>
<FONT color="green">042</FONT>     * &lt;p&gt;<a name="line.42"></a>
<FONT color="green">043</FONT>     * SYMMLQ is designed to solve the system (A - shift &amp;middot; I) &amp;middot; x = b,<a name="line.43"></a>
<FONT color="green">044</FONT>     * where shift is a specified scalar value. If shift and b are suitably chosen,<a name="line.44"></a>
<FONT color="green">045</FONT>     * the computed vector x may approximate an (unnormalized) eigenvector of A, as<a name="line.45"></a>
<FONT color="green">046</FONT>     * in the methods of inverse iteration and/or Rayleigh-quotient iteration.<a name="line.46"></a>
<FONT color="green">047</FONT>     * Again, the linear operator (A - shift &amp;middot; I) need not be positive<a name="line.47"></a>
<FONT color="green">048</FONT>     * definite (but &lt;em&gt;must&lt;/em&gt; be self-adjoint). The work per iteration is very<a name="line.48"></a>
<FONT color="green">049</FONT>     * slightly less if shift = 0.<a name="line.49"></a>
<FONT color="green">050</FONT>     * &lt;/p&gt;<a name="line.50"></a>
<FONT color="green">051</FONT>     * &lt;h3&gt;Preconditioning&lt;/h3&gt;<a name="line.51"></a>
<FONT color="green">052</FONT>     * &lt;p&gt;<a name="line.52"></a>
<FONT color="green">053</FONT>     * Preconditioning may reduce the number of iterations required. The solver may<a name="line.53"></a>
<FONT color="green">054</FONT>     * be provided with a positive definite preconditioner<a name="line.54"></a>
<FONT color="green">055</FONT>     * M = P&lt;sup&gt;T&lt;/sup&gt; &amp;middot; P<a name="line.55"></a>
<FONT color="green">056</FONT>     * that is known to approximate<a name="line.56"></a>
<FONT color="green">057</FONT>     * (A - shift &amp;middot; I)&lt;sup&gt;-1&lt;/sup&gt; in some sense, where matrix-vector<a name="line.57"></a>
<FONT color="green">058</FONT>     * products of the form M &amp;middot; y = x can be computed efficiently. Then<a name="line.58"></a>
<FONT color="green">059</FONT>     * SYMMLQ will implicitly solve the system of equations<a name="line.59"></a>
<FONT color="green">060</FONT>     * P &amp;middot; (A - shift &amp;middot; I) &amp;middot; P&lt;sup&gt;T&lt;/sup&gt; &amp;middot;<a name="line.60"></a>
<FONT color="green">061</FONT>     * x&lt;sub&gt;hat&lt;/sub&gt; = P &amp;middot; b, i.e.<a name="line.61"></a>
<FONT color="green">062</FONT>     * A&lt;sub&gt;hat&lt;/sub&gt; &amp;middot; x&lt;sub&gt;hat&lt;/sub&gt; = b&lt;sub&gt;hat&lt;/sub&gt;,<a name="line.62"></a>
<FONT color="green">063</FONT>     * where<a name="line.63"></a>
<FONT color="green">064</FONT>     * A&lt;sub&gt;hat&lt;/sub&gt; = P &amp;middot; (A - shift &amp;middot; I) &amp;middot; P&lt;sup&gt;T&lt;/sup&gt;,<a name="line.64"></a>
<FONT color="green">065</FONT>     * b&lt;sub&gt;hat&lt;/sub&gt; = P &amp;middot; b,<a name="line.65"></a>
<FONT color="green">066</FONT>     * and return the solution<a name="line.66"></a>
<FONT color="green">067</FONT>     * x = P&lt;sup&gt;T&lt;/sup&gt; &amp;middot; x&lt;sub&gt;hat&lt;/sub&gt;.<a name="line.67"></a>
<FONT color="green">068</FONT>     * The associated residual is<a name="line.68"></a>
<FONT color="green">069</FONT>     * r&lt;sub&gt;hat&lt;/sub&gt; = b&lt;sub&gt;hat&lt;/sub&gt; - A&lt;sub&gt;hat&lt;/sub&gt; &amp;middot; x&lt;sub&gt;hat&lt;/sub&gt;<a name="line.69"></a>
<FONT color="green">070</FONT>     *                 = P &amp;middot; [b - (A - shift &amp;middot; I) &amp;middot; x]<a name="line.70"></a>
<FONT color="green">071</FONT>     *                 = P &amp;middot; r.<a name="line.71"></a>
<FONT color="green">072</FONT>     * &lt;/p&gt;<a name="line.72"></a>
<FONT color="green">073</FONT>     * &lt;p&gt;<a name="line.73"></a>
<FONT color="green">074</FONT>     * In the case of preconditioning, the {@link IterativeLinearSolverEvent}s that<a name="line.74"></a>
<FONT color="green">075</FONT>     * this solver fires are such that<a name="line.75"></a>
<FONT color="green">076</FONT>     * {@link IterativeLinearSolverEvent#getNormOfResidual()} returns the norm of<a name="line.76"></a>
<FONT color="green">077</FONT>     * the &lt;em&gt;preconditioned&lt;/em&gt;, updated residual, ||P &amp;middot; r||, not the norm<a name="line.77"></a>
<FONT color="green">078</FONT>     * of the &lt;em&gt;true&lt;/em&gt; residual ||r||.<a name="line.78"></a>
<FONT color="green">079</FONT>     * &lt;/p&gt;<a name="line.79"></a>
<FONT color="green">080</FONT>     * &lt;h3&gt;&lt;a id="stopcrit"&gt;Default stopping criterion&lt;/a&gt;&lt;/h3&gt;<a name="line.80"></a>
<FONT color="green">081</FONT>     * &lt;p&gt;<a name="line.81"></a>
<FONT color="green">082</FONT>     * A default stopping criterion is implemented. The iterations stop when || rhat<a name="line.82"></a>
<FONT color="green">083</FONT>     * || &amp;le; &amp;delta; || Ahat || || xhat ||, where xhat is the current estimate of<a name="line.83"></a>
<FONT color="green">084</FONT>     * the solution of the transformed system, rhat the current estimate of the<a name="line.84"></a>
<FONT color="green">085</FONT>     * corresponding residual, and &amp;delta; a user-specified tolerance.<a name="line.85"></a>
<FONT color="green">086</FONT>     * &lt;/p&gt;<a name="line.86"></a>
<FONT color="green">087</FONT>     * &lt;h3&gt;Iteration count&lt;/h3&gt;<a name="line.87"></a>
<FONT color="green">088</FONT>     * &lt;p&gt;<a name="line.88"></a>
<FONT color="green">089</FONT>     * In the present context, an iteration should be understood as one evaluation<a name="line.89"></a>
<FONT color="green">090</FONT>     * of the matrix-vector product A &amp;middot; x. The initialization phase therefore<a name="line.90"></a>
<FONT color="green">091</FONT>     * counts as one iteration. If the user requires checks on the symmetry of A,<a name="line.91"></a>
<FONT color="green">092</FONT>     * this entails one further matrix-vector product in the initial phase. This<a name="line.92"></a>
<FONT color="green">093</FONT>     * further product is &lt;em&gt;not&lt;/em&gt; accounted for in the iteration count. In<a name="line.93"></a>
<FONT color="green">094</FONT>     * other words, the number of iterations required to reach convergence will be<a name="line.94"></a>
<FONT color="green">095</FONT>     * identical, whether checks have been required or not.<a name="line.95"></a>
<FONT color="green">096</FONT>     * &lt;/p&gt;<a name="line.96"></a>
<FONT color="green">097</FONT>     * &lt;p&gt;<a name="line.97"></a>
<FONT color="green">098</FONT>     * The present definition of the iteration count differs from that adopted in<a name="line.98"></a>
<FONT color="green">099</FONT>     * the original FOTRAN code, where the initialization phase was &lt;em&gt;not&lt;/em&gt;<a name="line.99"></a>
<FONT color="green">100</FONT>     * taken into account.<a name="line.100"></a>
<FONT color="green">101</FONT>     * &lt;/p&gt;<a name="line.101"></a>
<FONT color="green">102</FONT>     * &lt;h3&gt;&lt;a id="initguess"&gt;Initial guess of the solution&lt;/a&gt;&lt;/h3&gt;<a name="line.102"></a>
<FONT color="green">103</FONT>     * &lt;p&gt;<a name="line.103"></a>
<FONT color="green">104</FONT>     * The {@code x} parameter in<a name="line.104"></a>
<FONT color="green">105</FONT>     * &lt;ul&gt;<a name="line.105"></a>
<FONT color="green">106</FONT>     * &lt;li&gt;{@link #solve(RealLinearOperator, RealVector, RealVector)},&lt;/li&gt;<a name="line.106"></a>
<FONT color="green">107</FONT>     * &lt;li&gt;{@link #solve(RealLinearOperator, RealLinearOperator, RealVector, RealVector)}},&lt;/li&gt;<a name="line.107"></a>
<FONT color="green">108</FONT>     * &lt;li&gt;{@link #solveInPlace(RealLinearOperator, RealVector, RealVector)},&lt;/li&gt;<a name="line.108"></a>
<FONT color="green">109</FONT>     * &lt;li&gt;{@link #solveInPlace(RealLinearOperator, RealLinearOperator, RealVector, RealVector)},&lt;/li&gt;<a name="line.109"></a>
<FONT color="green">110</FONT>     * &lt;li&gt;{@link #solveInPlace(RealLinearOperator, RealLinearOperator, RealVector, RealVector, boolean, double)},&lt;/li&gt;<a name="line.110"></a>
<FONT color="green">111</FONT>     * &lt;/ul&gt;<a name="line.111"></a>
<FONT color="green">112</FONT>     * should not be considered as an initial guess, as it is set to zero in the<a name="line.112"></a>
<FONT color="green">113</FONT>     * initial phase. If x&lt;sub&gt;0&lt;/sub&gt; is known to be a good approximation to x, one<a name="line.113"></a>
<FONT color="green">114</FONT>     * should compute r&lt;sub&gt;0&lt;/sub&gt; = b - A &amp;middot; x, solve A &amp;middot; dx = r0,<a name="line.114"></a>
<FONT color="green">115</FONT>     * and set x = x&lt;sub&gt;0&lt;/sub&gt; + dx.<a name="line.115"></a>
<FONT color="green">116</FONT>     * &lt;/p&gt;<a name="line.116"></a>
<FONT color="green">117</FONT>     * &lt;h3&gt;&lt;a id="context"&gt;Exception context&lt;/a&gt;&lt;/h3&gt;<a name="line.117"></a>
<FONT color="green">118</FONT>     * &lt;p&gt;<a name="line.118"></a>
<FONT color="green">119</FONT>     * Besides standard {@link DimensionMismatchException}, this class might throw<a name="line.119"></a>
<FONT color="green">120</FONT>     * {@link NonSelfAdjointOperatorException} if the linear operator or the<a name="line.120"></a>
<FONT color="green">121</FONT>     * preconditioner are not symmetric. In this case, the {@link ExceptionContext}<a name="line.121"></a>
<FONT color="green">122</FONT>     * provides more information<a name="line.122"></a>
<FONT color="green">123</FONT>     * &lt;ul&gt;<a name="line.123"></a>
<FONT color="green">124</FONT>     * &lt;li&gt;key {@code "operator"} points to the offending linear operator, say L,&lt;/li&gt;<a name="line.124"></a>
<FONT color="green">125</FONT>     * &lt;li&gt;key {@code "vector1"} points to the first offending vector, say x,<a name="line.125"></a>
<FONT color="green">126</FONT>     * &lt;li&gt;key {@code "vector2"} points to the second offending vector, say y, such<a name="line.126"></a>
<FONT color="green">127</FONT>     * that x&lt;sup&gt;T&lt;/sup&gt; &amp;middot; L &amp;middot; y &amp;ne; y&lt;sup&gt;T&lt;/sup&gt; &amp;middot; L<a name="line.127"></a>
<FONT color="green">128</FONT>     * &amp;middot; x (within a certain accuracy).&lt;/li&gt;<a name="line.128"></a>
<FONT color="green">129</FONT>     * &lt;/ul&gt;<a name="line.129"></a>
<FONT color="green">130</FONT>     * &lt;/p&gt;<a name="line.130"></a>
<FONT color="green">131</FONT>     * &lt;p&gt;<a name="line.131"></a>
<FONT color="green">132</FONT>     * {@link NonPositiveDefiniteOperatorException} might also be thrown in case the<a name="line.132"></a>
<FONT color="green">133</FONT>     * preconditioner is not positive definite. The relevant keys to the<a name="line.133"></a>
<FONT color="green">134</FONT>     * {@link ExceptionContext} are<a name="line.134"></a>
<FONT color="green">135</FONT>     * &lt;ul&gt;<a name="line.135"></a>
<FONT color="green">136</FONT>     * &lt;li&gt;key {@code "operator"}, which points to the offending linear operator,<a name="line.136"></a>
<FONT color="green">137</FONT>     * say L,&lt;/li&gt;<a name="line.137"></a>
<FONT color="green">138</FONT>     * &lt;li&gt;key {@code "vector"}, which points to the offending vector, say x, such<a name="line.138"></a>
<FONT color="green">139</FONT>     * that x&lt;sup&gt;T&lt;/sup&gt; &amp;middot; L &amp;middot; x &lt; 0.&lt;/li&gt;<a name="line.139"></a>
<FONT color="green">140</FONT>     * &lt;/ul&gt;<a name="line.140"></a>
<FONT color="green">141</FONT>     * &lt;/p&gt;<a name="line.141"></a>
<FONT color="green">142</FONT>     * &lt;h3&gt;References&lt;/h3&gt;<a name="line.142"></a>
<FONT color="green">143</FONT>     * &lt;dl&gt;<a name="line.143"></a>
<FONT color="green">144</FONT>     * &lt;dt&gt;&lt;a id="PAIG1975"&gt;Paige and Saunders (1975)&lt;/a&gt;&lt;/dt&gt;<a name="line.144"></a>
<FONT color="green">145</FONT>     * &lt;dd&gt;C. C. Paige and M. A. Saunders, &lt;a<a name="line.145"></a>
<FONT color="green">146</FONT>     * href="http://www.stanford.edu/group/SOL/software/symmlq/PS75.pdf"&gt;&lt;em&gt;<a name="line.146"></a>
<FONT color="green">147</FONT>     * Solution of Sparse Indefinite Systems of Linear Equations&lt;/em&gt;&lt;/a&gt;, SIAM<a name="line.147"></a>
<FONT color="green">148</FONT>     * Journal on Numerical Analysis 12(4): 617-629, 1975&lt;/dd&gt;<a name="line.148"></a>
<FONT color="green">149</FONT>     * &lt;/dl&gt;<a name="line.149"></a>
<FONT color="green">150</FONT>     *<a name="line.150"></a>
<FONT color="green">151</FONT>     * @version $Id: SymmLQ.java 1416643 2012-12-03 19:37:14Z tn $<a name="line.151"></a>
<FONT color="green">152</FONT>     * @since 3.0<a name="line.152"></a>
<FONT color="green">153</FONT>     */<a name="line.153"></a>
<FONT color="green">154</FONT>    public class SymmLQ<a name="line.154"></a>
<FONT color="green">155</FONT>        extends PreconditionedIterativeLinearSolver {<a name="line.155"></a>
<FONT color="green">156</FONT>    <a name="line.156"></a>
<FONT color="green">157</FONT>        /*<a name="line.157"></a>
<FONT color="green">158</FONT>         * IMPLEMENTATION NOTES<a name="line.158"></a>
<FONT color="green">159</FONT>         * --------------------<a name="line.159"></a>
<FONT color="green">160</FONT>         * The implementation follows as closely as possible the notations of Paige<a name="line.160"></a>
<FONT color="green">161</FONT>         * and Saunders (1975). Attention must be paid to the fact that some<a name="line.161"></a>
<FONT color="green">162</FONT>         * quantities which are relevant to iteration k can only be computed in<a name="line.162"></a>
<FONT color="green">163</FONT>         * iteration (k+1). Therefore, minute attention must be paid to the index of<a name="line.163"></a>
<FONT color="green">164</FONT>         * each state variable of this algorithm.<a name="line.164"></a>
<FONT color="green">165</FONT>         *<a name="line.165"></a>
<FONT color="green">166</FONT>         * 1. Preconditioning<a name="line.166"></a>
<FONT color="green">167</FONT>         *    ---------------<a name="line.167"></a>
<FONT color="green">168</FONT>         * The Lanczos iterations associated with Ahat and bhat read<a name="line.168"></a>
<FONT color="green">169</FONT>         *   beta[1] = ||P * b||<a name="line.169"></a>
<FONT color="green">170</FONT>         *   v[1] = P * b / beta[1]<a name="line.170"></a>
<FONT color="green">171</FONT>         *   beta[k+1] * v[k+1] = Ahat * v[k] - alpha[k] * v[k] - beta[k] * v[k-1]<a name="line.171"></a>
<FONT color="green">172</FONT>         *                      = P * (A - shift * I) * P' * v[k] - alpha[k] * v[k]<a name="line.172"></a>
<FONT color="green">173</FONT>         *                        - beta[k] * v[k-1]<a name="line.173"></a>
<FONT color="green">174</FONT>         * Multiplying both sides by P', we get<a name="line.174"></a>
<FONT color="green">175</FONT>         *   beta[k+1] * (P' * v)[k+1] = M * (A - shift * I) * (P' * v)[k]<a name="line.175"></a>
<FONT color="green">176</FONT>         *                               - alpha[k] * (P' * v)[k]<a name="line.176"></a>
<FONT color="green">177</FONT>         *                               - beta[k] * (P' * v[k-1]),<a name="line.177"></a>
<FONT color="green">178</FONT>         * and<a name="line.178"></a>
<FONT color="green">179</FONT>         *   alpha[k+1] = v[k+1]' * Ahat * v[k+1]<a name="line.179"></a>
<FONT color="green">180</FONT>         *              = v[k+1]' * P * (A - shift * I) * P' * v[k+1]<a name="line.180"></a>
<FONT color="green">181</FONT>         *              = (P' * v)[k+1]' * (A - shift * I) * (P' * v)[k+1].<a name="line.181"></a>
<FONT color="green">182</FONT>         *<a name="line.182"></a>
<FONT color="green">183</FONT>         * In other words, the Lanczos iterations are unchanged, except for the fact<a name="line.183"></a>
<FONT color="green">184</FONT>         * that we really compute (P' * v) instead of v. It can easily be checked<a name="line.184"></a>
<FONT color="green">185</FONT>         * that all other formulas are unchanged. It must be noted that P is never<a name="line.185"></a>
<FONT color="green">186</FONT>         * explicitly used, only matrix-vector products involving are invoked.<a name="line.186"></a>
<FONT color="green">187</FONT>         *<a name="line.187"></a>
<FONT color="green">188</FONT>         * 2. Accounting for the shift parameter<a name="line.188"></a>
<FONT color="green">189</FONT>         *    ----------------------------------<a name="line.189"></a>
<FONT color="green">190</FONT>         * Is trivial: each time A.operate(x) is invoked, one must subtract shift * x<a name="line.190"></a>
<FONT color="green">191</FONT>         * to the result.<a name="line.191"></a>
<FONT color="green">192</FONT>         *<a name="line.192"></a>
<FONT color="green">193</FONT>         * 3. Accounting for the goodb flag<a name="line.193"></a>
<FONT color="green">194</FONT>         *    -----------------------------<a name="line.194"></a>
<FONT color="green">195</FONT>         * When goodb is set to true, the component of xL along b is computed<a name="line.195"></a>
<FONT color="green">196</FONT>         * separately. From Paige and Saunders (1975), equation (5.9), we have<a name="line.196"></a>
<FONT color="green">197</FONT>         *   wbar[k+1] = s[k] * wbar[k] - c[k] * v[k+1],<a name="line.197"></a>
<FONT color="green">198</FONT>         *   wbar[1] = v[1].<a name="line.198"></a>
<FONT color="green">199</FONT>         * Introducing wbar2[k] = wbar[k] - s[1] * ... * s[k-1] * v[1], it can<a name="line.199"></a>
<FONT color="green">200</FONT>         * easily be verified by induction that wbar2 follows the same recursive<a name="line.200"></a>
<FONT color="green">201</FONT>         * relation<a name="line.201"></a>
<FONT color="green">202</FONT>         *   wbar2[k+1] = s[k] * wbar2[k] - c[k] * v[k+1],<a name="line.202"></a>
<FONT color="green">203</FONT>         *   wbar2[1] = 0,<a name="line.203"></a>
<FONT color="green">204</FONT>         * and we then have<a name="line.204"></a>
<FONT color="green">205</FONT>         *   w[k] = c[k] * wbar2[k] + s[k] * v[k+1]<a name="line.205"></a>
<FONT color="green">206</FONT>         *          + s[1] * ... * s[k-1] * c[k] * v[1].<a name="line.206"></a>
<FONT color="green">207</FONT>         * Introducing w2[k] = w[k] - s[1] * ... * s[k-1] * c[k] * v[1], we find,<a name="line.207"></a>
<FONT color="green">208</FONT>         * from (5.10)<a name="line.208"></a>
<FONT color="green">209</FONT>         *   xL[k] = zeta[1] * w[1] + ... + zeta[k] * w[k]<a name="line.209"></a>
<FONT color="green">210</FONT>         *         = zeta[1] * w2[1] + ... + zeta[k] * w2[k]<a name="line.210"></a>
<FONT color="green">211</FONT>         *           + (s[1] * c[2] * zeta[2] + ...<a name="line.211"></a>
<FONT color="green">212</FONT>         *           + s[1] * ... * s[k-1] * c[k] * zeta[k]) * v[1]<a name="line.212"></a>
<FONT color="green">213</FONT>         *         = xL2[k] + bstep[k] * v[1],<a name="line.213"></a>
<FONT color="green">214</FONT>         * where xL2[k] is defined by<a name="line.214"></a>
<FONT color="green">215</FONT>         *   xL2[0] = 0,<a name="line.215"></a>
<FONT color="green">216</FONT>         *   xL2[k+1] = xL2[k] + zeta[k+1] * w2[k+1],<a name="line.216"></a>
<FONT color="green">217</FONT>         * and bstep is defined by<a name="line.217"></a>
<FONT color="green">218</FONT>         *   bstep[1] = 0,<a name="line.218"></a>
<FONT color="green">219</FONT>         *   bstep[k] = bstep[k-1] + s[1] * ... * s[k-1] * c[k] * zeta[k].<a name="line.219"></a>
<FONT color="green">220</FONT>         * We also have, from (5.11)<a name="line.220"></a>
<FONT color="green">221</FONT>         *   xC[k] = xL[k-1] + zbar[k] * wbar[k]<a name="line.221"></a>
<FONT color="green">222</FONT>         *         = xL2[k-1] + zbar[k] * wbar2[k]<a name="line.222"></a>
<FONT color="green">223</FONT>         *           + (bstep[k-1] + s[1] * ... * s[k-1] * zbar[k]) * v[1].<a name="line.223"></a>
<FONT color="green">224</FONT>         */<a name="line.224"></a>
<FONT color="green">225</FONT>    <a name="line.225"></a>
<FONT color="green">226</FONT>        /**<a name="line.226"></a>
<FONT color="green">227</FONT>         * &lt;p&gt;<a name="line.227"></a>
<FONT color="green">228</FONT>         * A simple container holding the non-final variables used in the<a name="line.228"></a>
<FONT color="green">229</FONT>         * iterations. Making the current state of the solver visible from the<a name="line.229"></a>
<FONT color="green">230</FONT>         * outside is necessary, because during the iterations, {@code x} does not<a name="line.230"></a>
<FONT color="green">231</FONT>         * &lt;em&gt;exactly&lt;/em&gt; hold the current estimate of the solution. Indeed,<a name="line.231"></a>
<FONT color="green">232</FONT>         * {@code x} needs in general to be moved from the LQ point to the CG point.<a name="line.232"></a>
<FONT color="green">233</FONT>         * Besides, additional upudates must be carried out in case {@code goodb} is<a name="line.233"></a>
<FONT color="green">234</FONT>         * set to {@code true}.<a name="line.234"></a>
<FONT color="green">235</FONT>         * &lt;/p&gt;<a name="line.235"></a>
<FONT color="green">236</FONT>         * &lt;p&gt;<a name="line.236"></a>
<FONT color="green">237</FONT>         * In all subsequent comments, the description of the state variables refer<a name="line.237"></a>
<FONT color="green">238</FONT>         * to their value after a call to {@link #update()}. In these comments, k is<a name="line.238"></a>
<FONT color="green">239</FONT>         * the current number of evaluations of matrix-vector products.<a name="line.239"></a>
<FONT color="green">240</FONT>         * &lt;/p&gt;<a name="line.240"></a>
<FONT color="green">241</FONT>         */<a name="line.241"></a>
<FONT color="green">242</FONT>        private static class State {<a name="line.242"></a>
<FONT color="green">243</FONT>            /** The cubic root of {@link #MACH_PREC}. */<a name="line.243"></a>
<FONT color="green">244</FONT>            static final double CBRT_MACH_PREC;<a name="line.244"></a>
<FONT color="green">245</FONT>    <a name="line.245"></a>
<FONT color="green">246</FONT>            /** The machine precision. */<a name="line.246"></a>
<FONT color="green">247</FONT>            static final double MACH_PREC;<a name="line.247"></a>
<FONT color="green">248</FONT>    <a name="line.248"></a>
<FONT color="green">249</FONT>            /** Reference to the linear operator. */<a name="line.249"></a>
<FONT color="green">250</FONT>            private final RealLinearOperator a;<a name="line.250"></a>
<FONT color="green">251</FONT>    <a name="line.251"></a>
<FONT color="green">252</FONT>            /** Reference to the right-hand side vector. */<a name="line.252"></a>
<FONT color="green">253</FONT>            private final RealVector b;<a name="line.253"></a>
<FONT color="green">254</FONT>    <a name="line.254"></a>
<FONT color="green">255</FONT>            /** {@code true} if symmetry of matrix and conditioner must be checked. */<a name="line.255"></a>
<FONT color="green">256</FONT>            private final boolean check;<a name="line.256"></a>
<FONT color="green">257</FONT>    <a name="line.257"></a>
<FONT color="green">258</FONT>            /**<a name="line.258"></a>
<FONT color="green">259</FONT>             * The value of the custom tolerance &amp;delta; for the default stopping<a name="line.259"></a>
<FONT color="green">260</FONT>             * criterion.<a name="line.260"></a>
<FONT color="green">261</FONT>             */<a name="line.261"></a>
<FONT color="green">262</FONT>            private final double delta;<a name="line.262"></a>
<FONT color="green">263</FONT>    <a name="line.263"></a>
<FONT color="green">264</FONT>            /** The value of beta[k+1]. */<a name="line.264"></a>
<FONT color="green">265</FONT>            private double beta;<a name="line.265"></a>
<FONT color="green">266</FONT>    <a name="line.266"></a>
<FONT color="green">267</FONT>            /** The value of beta[1]. */<a name="line.267"></a>
<FONT color="green">268</FONT>            private double beta1;<a name="line.268"></a>
<FONT color="green">269</FONT>    <a name="line.269"></a>
<FONT color="green">270</FONT>            /** The value of bstep[k-1]. */<a name="line.270"></a>
<FONT color="green">271</FONT>            private double bstep;<a name="line.271"></a>
<FONT color="green">272</FONT>    <a name="line.272"></a>
<FONT color="green">273</FONT>            /** The estimate of the norm of P * rC[k]. */<a name="line.273"></a>
<FONT color="green">274</FONT>            private double cgnorm;<a name="line.274"></a>
<FONT color="green">275</FONT>    <a name="line.275"></a>
<FONT color="green">276</FONT>            /** The value of dbar[k+1] = -beta[k+1] * c[k-1]. */<a name="line.276"></a>
<FONT color="green">277</FONT>            private double dbar;<a name="line.277"></a>
<FONT color="green">278</FONT>    <a name="line.278"></a>
<FONT color="green">279</FONT>            /**<a name="line.279"></a>
<FONT color="green">280</FONT>             * The value of gamma[k] * zeta[k]. Was called {@code rhs1} in the<a name="line.280"></a>
<FONT color="green">281</FONT>             * initial code.<a name="line.281"></a>
<FONT color="green">282</FONT>             */<a name="line.282"></a>
<FONT color="green">283</FONT>            private double gammaZeta;<a name="line.283"></a>
<FONT color="green">284</FONT>    <a name="line.284"></a>
<FONT color="green">285</FONT>            /** The value of gbar[k]. */<a name="line.285"></a>
<FONT color="green">286</FONT>            private double gbar;<a name="line.286"></a>
<FONT color="green">287</FONT>    <a name="line.287"></a>
<FONT color="green">288</FONT>            /** The value of max(|alpha[1]|, gamma[1], ..., gamma[k-1]). */<a name="line.288"></a>
<FONT color="green">289</FONT>            private double gmax;<a name="line.289"></a>
<FONT color="green">290</FONT>    <a name="line.290"></a>
<FONT color="green">291</FONT>            /** The value of min(|alpha[1]|, gamma[1], ..., gamma[k-1]). */<a name="line.291"></a>
<FONT color="green">292</FONT>            private double gmin;<a name="line.292"></a>
<FONT color="green">293</FONT>    <a name="line.293"></a>
<FONT color="green">294</FONT>            /** Copy of the {@code goodb} parameter. */<a name="line.294"></a>
<FONT color="green">295</FONT>            private final boolean goodb;<a name="line.295"></a>
<FONT color="green">296</FONT>    <a name="line.296"></a>
<FONT color="green">297</FONT>            /** {@code true} if the default convergence criterion is verified. */<a name="line.297"></a>
<FONT color="green">298</FONT>            private boolean hasConverged;<a name="line.298"></a>
<FONT color="green">299</FONT>    <a name="line.299"></a>
<FONT color="green">300</FONT>            /** The estimate of the norm of P * rL[k-1]. */<a name="line.300"></a>
<FONT color="green">301</FONT>            private double lqnorm;<a name="line.301"></a>
<FONT color="green">302</FONT>    <a name="line.302"></a>
<FONT color="green">303</FONT>            /** Reference to the preconditioner, M. */<a name="line.303"></a>
<FONT color="green">304</FONT>            private final RealLinearOperator m;<a name="line.304"></a>
<FONT color="green">305</FONT>    <a name="line.305"></a>
<FONT color="green">306</FONT>            /**<a name="line.306"></a>
<FONT color="green">307</FONT>             * The value of (-eps[k+1] * zeta[k-1]). Was called {@code rhs2} in the<a name="line.307"></a>
<FONT color="green">308</FONT>             * initial code.<a name="line.308"></a>
<FONT color="green">309</FONT>             */<a name="line.309"></a>
<FONT color="green">310</FONT>            private double minusEpsZeta;<a name="line.310"></a>
<FONT color="green">311</FONT>    <a name="line.311"></a>
<FONT color="green">312</FONT>            /** The value of M * b. */<a name="line.312"></a>
<FONT color="green">313</FONT>            private final RealVector mb;<a name="line.313"></a>
<FONT color="green">314</FONT>    <a name="line.314"></a>
<FONT color="green">315</FONT>            /** The value of beta[k]. */<a name="line.315"></a>
<FONT color="green">316</FONT>            private double oldb;<a name="line.316"></a>
<FONT color="green">317</FONT>    <a name="line.317"></a>
<FONT color="green">318</FONT>            /** The value of beta[k] * M^(-1) * P' * v[k]. */<a name="line.318"></a>
<FONT color="green">319</FONT>            private RealVector r1;<a name="line.319"></a>
<FONT color="green">320</FONT>    <a name="line.320"></a>
<FONT color="green">321</FONT>            /** The value of beta[k+1] * M^(-1) * P' * v[k+1]. */<a name="line.321"></a>
<FONT color="green">322</FONT>            private RealVector r2;<a name="line.322"></a>
<FONT color="green">323</FONT>    <a name="line.323"></a>
<FONT color="green">324</FONT>            /**<a name="line.324"></a>
<FONT color="green">325</FONT>             * The value of the updated, preconditioned residual P * r. This value is<a name="line.325"></a>
<FONT color="green">326</FONT>             * given by {@code min(}{@link #cgnorm}{@code , }{@link #lqnorm}{@code )}.<a name="line.326"></a>
<FONT color="green">327</FONT>             */<a name="line.327"></a>
<FONT color="green">328</FONT>            private double rnorm;<a name="line.328"></a>
<FONT color="green">329</FONT>    <a name="line.329"></a>
<FONT color="green">330</FONT>            /** Copy of the {@code shift} parameter. */<a name="line.330"></a>
<FONT color="green">331</FONT>            private final double shift;<a name="line.331"></a>
<FONT color="green">332</FONT>    <a name="line.332"></a>
<FONT color="green">333</FONT>            /** The value of s[1] * ... * s[k-1]. */<a name="line.333"></a>
<FONT color="green">334</FONT>            private double snprod;<a name="line.334"></a>
<FONT color="green">335</FONT>    <a name="line.335"></a>
<FONT color="green">336</FONT>            /**<a name="line.336"></a>
<FONT color="green">337</FONT>             * An estimate of the square of the norm of A * V[k], based on Paige and<a name="line.337"></a>
<FONT color="green">338</FONT>             * Saunders (1975), equation (3.3).<a name="line.338"></a>
<FONT color="green">339</FONT>             */<a name="line.339"></a>
<FONT color="green">340</FONT>            private double tnorm;<a name="line.340"></a>
<FONT color="green">341</FONT>    <a name="line.341"></a>
<FONT color="green">342</FONT>            /**<a name="line.342"></a>
<FONT color="green">343</FONT>             * The value of P' * wbar[k] or P' * (wbar[k] - s[1] * ... * s[k-1] *<a name="line.343"></a>
<FONT color="green">344</FONT>             * v[1]) if {@code goodb} is {@code true}. Was called {@code w} in the<a name="line.344"></a>
<FONT color="green">345</FONT>             * initial code.<a name="line.345"></a>
<FONT color="green">346</FONT>             */<a name="line.346"></a>
<FONT color="green">347</FONT>            private RealVector wbar;<a name="line.347"></a>
<FONT color="green">348</FONT>    <a name="line.348"></a>
<FONT color="green">349</FONT>            /**<a name="line.349"></a>
<FONT color="green">350</FONT>             * A reference to the vector to be updated with the solution. Contains<a name="line.350"></a>
<FONT color="green">351</FONT>             * the value of xL[k-1] if {@code goodb} is {@code false}, (xL[k-1] -<a name="line.351"></a>
<FONT color="green">352</FONT>             * bstep[k-1] * v[1]) otherwise.<a name="line.352"></a>
<FONT color="green">353</FONT>             */<a name="line.353"></a>
<FONT color="green">354</FONT>            private final RealVector xL;<a name="line.354"></a>
<FONT color="green">355</FONT>    <a name="line.355"></a>
<FONT color="green">356</FONT>            /** The value of beta[k+1] * P' * v[k+1]. */<a name="line.356"></a>
<FONT color="green">357</FONT>            private RealVector y;<a name="line.357"></a>
<FONT color="green">358</FONT>    <a name="line.358"></a>
<FONT color="green">359</FONT>            /** The value of zeta[1]^2 + ... + zeta[k-1]^2. */<a name="line.359"></a>
<FONT color="green">360</FONT>            private double ynorm2;<a name="line.360"></a>
<FONT color="green">361</FONT>    <a name="line.361"></a>
<FONT color="green">362</FONT>            /** The value of {@code b == 0} (exact floating-point equality). */<a name="line.362"></a>
<FONT color="green">363</FONT>            private boolean bIsNull;<a name="line.363"></a>
<FONT color="green">364</FONT>    <a name="line.364"></a>
<FONT color="green">365</FONT>            static {<a name="line.365"></a>
<FONT color="green">366</FONT>                MACH_PREC = FastMath.ulp(1.);<a name="line.366"></a>
<FONT color="green">367</FONT>                CBRT_MACH_PREC = FastMath.cbrt(MACH_PREC);<a name="line.367"></a>
<FONT color="green">368</FONT>            }<a name="line.368"></a>
<FONT color="green">369</FONT>    <a name="line.369"></a>
<FONT color="green">370</FONT>            /**<a name="line.370"></a>
<FONT color="green">371</FONT>             * Creates and inits to k = 1 a new instance of this class.<a name="line.371"></a>
<FONT color="green">372</FONT>             *<a name="line.372"></a>
<FONT color="green">373</FONT>             * @param a the linear operator A of the system<a name="line.373"></a>
<FONT color="green">374</FONT>             * @param m the preconditioner, M (can be {@code null})<a name="line.374"></a>
<FONT color="green">375</FONT>             * @param b the right-hand side vector<a name="line.375"></a>
<FONT color="green">376</FONT>             * @param goodb usually {@code false}, except if {@code x} is expected<a name="line.376"></a>
<FONT color="green">377</FONT>             * to contain a large multiple of {@code b}<a name="line.377"></a>
<FONT color="green">378</FONT>             * @param shift the amount to be subtracted to all diagonal elements of<a name="line.378"></a>
<FONT color="green">379</FONT>             * A<a name="line.379"></a>
<FONT color="green">380</FONT>             * @param delta the &amp;delta; parameter for the default stopping criterion<a name="line.380"></a>
<FONT color="green">381</FONT>             * @param check {@code true} if self-adjointedness of both matrix and<a name="line.381"></a>
<FONT color="green">382</FONT>             * preconditioner should be checked<a name="line.382"></a>
<FONT color="green">383</FONT>             */<a name="line.383"></a>
<FONT color="green">384</FONT>            public State(final RealLinearOperator a,<a name="line.384"></a>
<FONT color="green">385</FONT>                final RealLinearOperator m,<a name="line.385"></a>
<FONT color="green">386</FONT>                final RealVector b,<a name="line.386"></a>
<FONT color="green">387</FONT>                final boolean goodb,<a name="line.387"></a>
<FONT color="green">388</FONT>                final double shift,<a name="line.388"></a>
<FONT color="green">389</FONT>                final double delta,<a name="line.389"></a>
<FONT color="green">390</FONT>                final boolean check) {<a name="line.390"></a>
<FONT color="green">391</FONT>                this.a = a;<a name="line.391"></a>
<FONT color="green">392</FONT>                this.m = m;<a name="line.392"></a>
<FONT color="green">393</FONT>                this.b = b;<a name="line.393"></a>
<FONT color="green">394</FONT>                this.xL = new ArrayRealVector(b.getDimension());<a name="line.394"></a>
<FONT color="green">395</FONT>                this.goodb = goodb;<a name="line.395"></a>
<FONT color="green">396</FONT>                this.shift = shift;<a name="line.396"></a>
<FONT color="green">397</FONT>                this.mb = m == null ? b : m.operate(b);<a name="line.397"></a>
<FONT color="green">398</FONT>                this.hasConverged = false;<a name="line.398"></a>
<FONT color="green">399</FONT>                this.check = check;<a name="line.399"></a>
<FONT color="green">400</FONT>                this.delta = delta;<a name="line.400"></a>
<FONT color="green">401</FONT>            }<a name="line.401"></a>
<FONT color="green">402</FONT>    <a name="line.402"></a>
<FONT color="green">403</FONT>            /**<a name="line.403"></a>
<FONT color="green">404</FONT>             * Performs a symmetry check on the specified linear operator, and throws an<a name="line.404"></a>
<FONT color="green">405</FONT>             * exception in case this check fails. Given a linear operator L, and a<a name="line.405"></a>
<FONT color="green">406</FONT>             * vector x, this method checks that<a name="line.406"></a>
<FONT color="green">407</FONT>             * x' &amp;middot; L &amp;middot; y = y' &amp;middot; L &amp;middot; x<a name="line.407"></a>
<FONT color="green">408</FONT>             * (within a given accuracy), where y = L &amp;middot; x.<a name="line.408"></a>
<FONT color="green">409</FONT>             *<a name="line.409"></a>
<FONT color="green">410</FONT>             * @param l the linear operator L<a name="line.410"></a>
<FONT color="green">411</FONT>             * @param x the candidate vector x<a name="line.411"></a>
<FONT color="green">412</FONT>             * @param y the candidate vector y = L &amp;middot; x<a name="line.412"></a>
<FONT color="green">413</FONT>             * @param z the vector z = L &amp;middot; y<a name="line.413"></a>
<FONT color="green">414</FONT>             * @throws NonSelfAdjointOperatorException when the test fails<a name="line.414"></a>
<FONT color="green">415</FONT>             */<a name="line.415"></a>
<FONT color="green">416</FONT>            private static void checkSymmetry(final RealLinearOperator l,<a name="line.416"></a>
<FONT color="green">417</FONT>                final RealVector x, final RealVector y, final RealVector z)<a name="line.417"></a>
<FONT color="green">418</FONT>                throws NonSelfAdjointOperatorException {<a name="line.418"></a>
<FONT color="green">419</FONT>                final double s = y.dotProduct(y);<a name="line.419"></a>
<FONT color="green">420</FONT>                final double t = x.dotProduct(z);<a name="line.420"></a>
<FONT color="green">421</FONT>                final double epsa = (s + MACH_PREC) * CBRT_MACH_PREC;<a name="line.421"></a>
<FONT color="green">422</FONT>                if (FastMath.abs(s - t) &gt; epsa) {<a name="line.422"></a>
<FONT color="green">423</FONT>                    final NonSelfAdjointOperatorException e;<a name="line.423"></a>
<FONT color="green">424</FONT>                    e = new NonSelfAdjointOperatorException();<a name="line.424"></a>
<FONT color="green">425</FONT>                    final ExceptionContext context = e.getContext();<a name="line.425"></a>
<FONT color="green">426</FONT>                    context.setValue(SymmLQ.OPERATOR, l);<a name="line.426"></a>
<FONT color="green">427</FONT>                    context.setValue(SymmLQ.VECTOR1, x);<a name="line.427"></a>
<FONT color="green">428</FONT>                    context.setValue(SymmLQ.VECTOR2, y);<a name="line.428"></a>
<FONT color="green">429</FONT>                    context.setValue(SymmLQ.THRESHOLD, Double.valueOf(epsa));<a name="line.429"></a>
<FONT color="green">430</FONT>                    throw e;<a name="line.430"></a>
<FONT color="green">431</FONT>                }<a name="line.431"></a>
<FONT color="green">432</FONT>            }<a name="line.432"></a>
<FONT color="green">433</FONT>    <a name="line.433"></a>
<FONT color="green">434</FONT>            /**<a name="line.434"></a>
<FONT color="green">435</FONT>             * Throws a new {@link NonPositiveDefiniteOperatorException} with<a name="line.435"></a>
<FONT color="green">436</FONT>             * appropriate context.<a name="line.436"></a>
<FONT color="green">437</FONT>             *<a name="line.437"></a>
<FONT color="green">438</FONT>             * @param l the offending linear operator<a name="line.438"></a>
<FONT color="green">439</FONT>             * @param v the offending vector<a name="line.439"></a>
<FONT color="green">440</FONT>             * @throws NonPositiveDefiniteOperatorException in any circumstances<a name="line.440"></a>
<FONT color="green">441</FONT>             */<a name="line.441"></a>
<FONT color="green">442</FONT>            private static void throwNPDLOException(final RealLinearOperator l,<a name="line.442"></a>
<FONT color="green">443</FONT>                final RealVector v) throws NonPositiveDefiniteOperatorException {<a name="line.443"></a>
<FONT color="green">444</FONT>                final NonPositiveDefiniteOperatorException e;<a name="line.444"></a>
<FONT color="green">445</FONT>                e = new NonPositiveDefiniteOperatorException();<a name="line.445"></a>
<FONT color="green">446</FONT>                final ExceptionContext context = e.getContext();<a name="line.446"></a>
<FONT color="green">447</FONT>                context.setValue(OPERATOR, l);<a name="line.447"></a>
<FONT color="green">448</FONT>                context.setValue(VECTOR, v);<a name="line.448"></a>
<FONT color="green">449</FONT>                throw e;<a name="line.449"></a>
<FONT color="green">450</FONT>            }<a name="line.450"></a>
<FONT color="green">451</FONT>    <a name="line.451"></a>
<FONT color="green">452</FONT>            /**<a name="line.452"></a>
<FONT color="green">453</FONT>             * A clone of the BLAS {@code DAXPY} function, which carries out the<a name="line.453"></a>
<FONT color="green">454</FONT>             * operation y &amp;larr; a &amp;middot; x + y. This is for internal use only: no<a name="line.454"></a>
<FONT color="green">455</FONT>             * dimension checks are provided.<a name="line.455"></a>
<FONT color="green">456</FONT>             *<a name="line.456"></a>
<FONT color="green">457</FONT>             * @param a the scalar by which {@code x} is to be multiplied<a name="line.457"></a>
<FONT color="green">458</FONT>             * @param x the vector to be added to {@code y}<a name="line.458"></a>
<FONT color="green">459</FONT>             * @param y the vector to be incremented<a name="line.459"></a>
<FONT color="green">460</FONT>             */<a name="line.460"></a>
<FONT color="green">461</FONT>            private static void daxpy(final double a, final RealVector x,<a name="line.461"></a>
<FONT color="green">462</FONT>                final RealVector y) {<a name="line.462"></a>
<FONT color="green">463</FONT>                final int n = x.getDimension();<a name="line.463"></a>
<FONT color="green">464</FONT>                for (int i = 0; i &lt; n; i++) {<a name="line.464"></a>
<FONT color="green">465</FONT>                    y.setEntry(i, a * x.getEntry(i) + y.getEntry(i));<a name="line.465"></a>
<FONT color="green">466</FONT>                }<a name="line.466"></a>
<FONT color="green">467</FONT>            }<a name="line.467"></a>
<FONT color="green">468</FONT>    <a name="line.468"></a>
<FONT color="green">469</FONT>            /**<a name="line.469"></a>
<FONT color="green">470</FONT>             * A BLAS-like function, for the operation z &amp;larr; a &amp;middot; x + b<a name="line.470"></a>
<FONT color="green">471</FONT>             * &amp;middot; y + z. This is for internal use only: no dimension checks are<a name="line.471"></a>
<FONT color="green">472</FONT>             * provided.<a name="line.472"></a>
<FONT color="green">473</FONT>             *<a name="line.473"></a>
<FONT color="green">474</FONT>             * @param a the scalar by which {@code x} is to be multiplied<a name="line.474"></a>
<FONT color="green">475</FONT>             * @param x the first vector to be added to {@code z}<a name="line.475"></a>
<FONT color="green">476</FONT>             * @param b the scalar by which {@code y} is to be multiplied<a name="line.476"></a>
<FONT color="green">477</FONT>             * @param y the second vector to be added to {@code z}<a name="line.477"></a>
<FONT color="green">478</FONT>             * @param z the vector to be incremented<a name="line.478"></a>
<FONT color="green">479</FONT>             */<a name="line.479"></a>
<FONT color="green">480</FONT>            private static void daxpbypz(final double a, final RealVector x,<a name="line.480"></a>
<FONT color="green">481</FONT>                final double b, final RealVector y, final RealVector z) {<a name="line.481"></a>
<FONT color="green">482</FONT>                final int n = z.getDimension();<a name="line.482"></a>
<FONT color="green">483</FONT>                for (int i = 0; i &lt; n; i++) {<a name="line.483"></a>
<FONT color="green">484</FONT>                    final double zi;<a name="line.484"></a>
<FONT color="green">485</FONT>                    zi = a * x.getEntry(i) + b * y.getEntry(i) + z.getEntry(i);<a name="line.485"></a>
<FONT color="green">486</FONT>                    z.setEntry(i, zi);<a name="line.486"></a>
<FONT color="green">487</FONT>                }<a name="line.487"></a>
<FONT color="green">488</FONT>            }<a name="line.488"></a>
<FONT color="green">489</FONT>    <a name="line.489"></a>
<FONT color="green">490</FONT>            /**<a name="line.490"></a>
<FONT color="green">491</FONT>             * &lt;p&gt;<a name="line.491"></a>
<FONT color="green">492</FONT>             * Move to the CG point if it seems better. In this version of SYMMLQ,<a name="line.492"></a>
<FONT color="green">493</FONT>             * the convergence tests involve only cgnorm, so we're unlikely to stop<a name="line.493"></a>
<FONT color="green">494</FONT>             * at an LQ point, except if the iteration limit interferes.<a name="line.494"></a>
<FONT color="green">495</FONT>             * &lt;/p&gt;<a name="line.495"></a>
<FONT color="green">496</FONT>             * &lt;p&gt;<a name="line.496"></a>
<FONT color="green">497</FONT>             * Additional upudates are also carried out in case {@code goodb} is set<a name="line.497"></a>
<FONT color="green">498</FONT>             * to {@code true}.<a name="line.498"></a>
<FONT color="green">499</FONT>             * &lt;/p&gt;<a name="line.499"></a>
<FONT color="green">500</FONT>             *<a name="line.500"></a>
<FONT color="green">501</FONT>             * @param x the vector to be updated with the refined value of xL<a name="line.501"></a>
<FONT color="green">502</FONT>             */<a name="line.502"></a>
<FONT color="green">503</FONT>             void refineSolution(final RealVector x) {<a name="line.503"></a>
<FONT color="green">504</FONT>                final int n = this.xL.getDimension();<a name="line.504"></a>
<FONT color="green">505</FONT>                if (lqnorm &lt; cgnorm) {<a name="line.505"></a>
<FONT color="green">506</FONT>                    if (!goodb) {<a name="line.506"></a>
<FONT color="green">507</FONT>                        x.setSubVector(0, this.xL);<a name="line.507"></a>
<FONT color="green">508</FONT>                    } else {<a name="line.508"></a>
<FONT color="green">509</FONT>                        final double step = bstep / beta1;<a name="line.509"></a>
<FONT color="green">510</FONT>                        for (int i = 0; i &lt; n; i++) {<a name="line.510"></a>
<FONT color="green">511</FONT>                            final double bi = mb.getEntry(i);<a name="line.511"></a>
<FONT color="green">512</FONT>                            final double xi = this.xL.getEntry(i);<a name="line.512"></a>
<FONT color="green">513</FONT>                            x.setEntry(i, xi + step * bi);<a name="line.513"></a>
<FONT color="green">514</FONT>                        }<a name="line.514"></a>
<FONT color="green">515</FONT>                    }<a name="line.515"></a>
<FONT color="green">516</FONT>                } else {<a name="line.516"></a>
<FONT color="green">517</FONT>                    final double anorm = FastMath.sqrt(tnorm);<a name="line.517"></a>
<FONT color="green">518</FONT>                    final double diag = gbar == 0. ? anorm * MACH_PREC : gbar;<a name="line.518"></a>
<FONT color="green">519</FONT>                    final double zbar = gammaZeta / diag;<a name="line.519"></a>
<FONT color="green">520</FONT>                    final double step = (bstep + snprod * zbar) / beta1;<a name="line.520"></a>
<FONT color="green">521</FONT>                    // ynorm = FastMath.sqrt(ynorm2 + zbar * zbar);<a name="line.521"></a>
<FONT color="green">522</FONT>                    if (!goodb) {<a name="line.522"></a>
<FONT color="green">523</FONT>                        for (int i = 0; i &lt; n; i++) {<a name="line.523"></a>
<FONT color="green">524</FONT>                            final double xi = this.xL.getEntry(i);<a name="line.524"></a>
<FONT color="green">525</FONT>                            final double wi = wbar.getEntry(i);<a name="line.525"></a>
<FONT color="green">526</FONT>                            x.setEntry(i, xi + zbar * wi);<a name="line.526"></a>
<FONT color="green">527</FONT>                        }<a name="line.527"></a>
<FONT color="green">528</FONT>                    } else {<a name="line.528"></a>
<FONT color="green">529</FONT>                        for (int i = 0; i &lt; n; i++) {<a name="line.529"></a>
<FONT color="green">530</FONT>                            final double xi = this.xL.getEntry(i);<a name="line.530"></a>
<FONT color="green">531</FONT>                            final double wi = wbar.getEntry(i);<a name="line.531"></a>
<FONT color="green">532</FONT>                            final double bi = mb.getEntry(i);<a name="line.532"></a>
<FONT color="green">533</FONT>                            x.setEntry(i, xi + zbar * wi + step * bi);<a name="line.533"></a>
<FONT color="green">534</FONT>                        }<a name="line.534"></a>
<FONT color="green">535</FONT>                    }<a name="line.535"></a>
<FONT color="green">536</FONT>                }<a name="line.536"></a>
<FONT color="green">537</FONT>            }<a name="line.537"></a>
<FONT color="green">538</FONT>    <a name="line.538"></a>
<FONT color="green">539</FONT>            /**<a name="line.539"></a>
<FONT color="green">540</FONT>             * Performs the initial phase of the SYMMLQ algorithm. On return, the<a name="line.540"></a>
<FONT color="green">541</FONT>             * value of the state variables of {@code this} object correspond to k =<a name="line.541"></a>
<FONT color="green">542</FONT>             * 1.<a name="line.542"></a>
<FONT color="green">543</FONT>             */<a name="line.543"></a>
<FONT color="green">544</FONT>             void init() {<a name="line.544"></a>
<FONT color="green">545</FONT>                this.xL.set(0.);<a name="line.545"></a>
<FONT color="green">546</FONT>                /*<a name="line.546"></a>
<FONT color="green">547</FONT>                 * Set up y for the first Lanczos vector. y and beta1 will be zero<a name="line.547"></a>
<FONT color="green">548</FONT>                 * if b = 0.<a name="line.548"></a>
<FONT color="green">549</FONT>                 */<a name="line.549"></a>
<FONT color="green">550</FONT>                this.r1 = this.b.copy();<a name="line.550"></a>
<FONT color="green">551</FONT>                this.y = this.m == null ? this.b.copy() : this.m.operate(this.r1);<a name="line.551"></a>
<FONT color="green">552</FONT>                if ((this.m != null) &amp;&amp; this.check) {<a name="line.552"></a>
<FONT color="green">553</FONT>                    checkSymmetry(this.m, this.r1, this.y, this.m.operate(this.y));<a name="line.553"></a>
<FONT color="green">554</FONT>                }<a name="line.554"></a>
<FONT color="green">555</FONT>    <a name="line.555"></a>
<FONT color="green">556</FONT>                this.beta1 = this.r1.dotProduct(this.y);<a name="line.556"></a>
<FONT color="green">557</FONT>                if (this.beta1 &lt; 0.) {<a name="line.557"></a>
<FONT color="green">558</FONT>                    throwNPDLOException(this.m, this.y);<a name="line.558"></a>
<FONT color="green">559</FONT>                }<a name="line.559"></a>
<FONT color="green">560</FONT>                if (this.beta1 == 0.) {<a name="line.560"></a>
<FONT color="green">561</FONT>                    /* If b = 0 exactly, stop with x = 0. */<a name="line.561"></a>
<FONT color="green">562</FONT>                    this.bIsNull = true;<a name="line.562"></a>
<FONT color="green">563</FONT>                    return;<a name="line.563"></a>
<FONT color="green">564</FONT>                }<a name="line.564"></a>
<FONT color="green">565</FONT>                this.bIsNull = false;<a name="line.565"></a>
<FONT color="green">566</FONT>                this.beta1 = FastMath.sqrt(this.beta1);<a name="line.566"></a>
<FONT color="green">567</FONT>                /* At this point<a name="line.567"></a>
<FONT color="green">568</FONT>                 *   r1 = b,<a name="line.568"></a>
<FONT color="green">569</FONT>                 *   y = M * b,<a name="line.569"></a>
<FONT color="green">570</FONT>                 *   beta1 = beta[1].<a name="line.570"></a>
<FONT color="green">571</FONT>                 */<a name="line.571"></a>
<FONT color="green">572</FONT>                final RealVector v = this.y.mapMultiply(1. / this.beta1);<a name="line.572"></a>
<FONT color="green">573</FONT>                this.y = this.a.operate(v);<a name="line.573"></a>
<FONT color="green">574</FONT>                if (this.check) {<a name="line.574"></a>
<FONT color="green">575</FONT>                    checkSymmetry(this.a, v, this.y, this.a.operate(this.y));<a name="line.575"></a>
<FONT color="green">576</FONT>                }<a name="line.576"></a>
<FONT color="green">577</FONT>                /*<a name="line.577"></a>
<FONT color="green">578</FONT>                 * Set up y for the second Lanczos vector. y and beta will be zero<a name="line.578"></a>
<FONT color="green">579</FONT>                 * or very small if b is an eigenvector.<a name="line.579"></a>
<FONT color="green">580</FONT>                 */<a name="line.580"></a>
<FONT color="green">581</FONT>                daxpy(-this.shift, v, this.y);<a name="line.581"></a>
<FONT color="green">582</FONT>                final double alpha = v.dotProduct(this.y);<a name="line.582"></a>
<FONT color="green">583</FONT>                daxpy(-alpha / this.beta1, this.r1, this.y);<a name="line.583"></a>
<FONT color="green">584</FONT>                /*<a name="line.584"></a>
<FONT color="green">585</FONT>                 * At this point<a name="line.585"></a>
<FONT color="green">586</FONT>                 *   alpha = alpha[1]<a name="line.586"></a>
<FONT color="green">587</FONT>                 *   y     = beta[2] * M^(-1) * P' * v[2]<a name="line.587"></a>
<FONT color="green">588</FONT>                 */<a name="line.588"></a>
<FONT color="green">589</FONT>                /* Make sure r2 will be orthogonal to the first v. */<a name="line.589"></a>
<FONT color="green">590</FONT>                final double vty = v.dotProduct(this.y);<a name="line.590"></a>
<FONT color="green">591</FONT>                final double vtv = v.dotProduct(v);<a name="line.591"></a>
<FONT color="green">592</FONT>                daxpy(-vty / vtv, v, this.y);<a name="line.592"></a>
<FONT color="green">593</FONT>                this.r2 = this.y.copy();<a name="line.593"></a>
<FONT color="green">594</FONT>                if (this.m != null) {<a name="line.594"></a>
<FONT color="green">595</FONT>                    this.y = this.m.operate(this.r2);<a name="line.595"></a>
<FONT color="green">596</FONT>                }<a name="line.596"></a>
<FONT color="green">597</FONT>                this.oldb = this.beta1;<a name="line.597"></a>
<FONT color="green">598</FONT>                this.beta = this.r2.dotProduct(this.y);<a name="line.598"></a>
<FONT color="green">599</FONT>                if (this.beta &lt; 0.) {<a name="line.599"></a>
<FONT color="green">600</FONT>                    throwNPDLOException(this.m, this.y);<a name="line.600"></a>
<FONT color="green">601</FONT>                }<a name="line.601"></a>
<FONT color="green">602</FONT>                this.beta = FastMath.sqrt(this.beta);<a name="line.602"></a>
<FONT color="green">603</FONT>                /*<a name="line.603"></a>
<FONT color="green">604</FONT>                 * At this point<a name="line.604"></a>
<FONT color="green">605</FONT>                 *   oldb = beta[1]<a name="line.605"></a>
<FONT color="green">606</FONT>                 *   beta = beta[2]<a name="line.606"></a>
<FONT color="green">607</FONT>                 *   y  = beta[2] * P' * v[2]<a name="line.607"></a>
<FONT color="green">608</FONT>                 *   r2 = beta[2] * M^(-1) * P' * v[2]<a name="line.608"></a>
<FONT color="green">609</FONT>                 */<a name="line.609"></a>
<FONT color="green">610</FONT>                this.cgnorm = this.beta1;<a name="line.610"></a>
<FONT color="green">611</FONT>                this.gbar = alpha;<a name="line.611"></a>
<FONT color="green">612</FONT>                this.dbar = this.beta;<a name="line.612"></a>
<FONT color="green">613</FONT>                this.gammaZeta = this.beta1;<a name="line.613"></a>
<FONT color="green">614</FONT>                this.minusEpsZeta = 0.;<a name="line.614"></a>
<FONT color="green">615</FONT>                this.bstep = 0.;<a name="line.615"></a>
<FONT color="green">616</FONT>                this.snprod = 1.;<a name="line.616"></a>
<FONT color="green">617</FONT>                this.tnorm = alpha * alpha + this.beta * this.beta;<a name="line.617"></a>
<FONT color="green">618</FONT>                this.ynorm2 = 0.;<a name="line.618"></a>
<FONT color="green">619</FONT>                this.gmax = FastMath.abs(alpha) + MACH_PREC;<a name="line.619"></a>
<FONT color="green">620</FONT>                this.gmin = this.gmax;<a name="line.620"></a>
<FONT color="green">621</FONT>    <a name="line.621"></a>
<FONT color="green">622</FONT>                if (this.goodb) {<a name="line.622"></a>
<FONT color="green">623</FONT>                    this.wbar = new ArrayRealVector(this.a.getRowDimension());<a name="line.623"></a>
<FONT color="green">624</FONT>                    this.wbar.set(0.);<a name="line.624"></a>
<FONT color="green">625</FONT>                } else {<a name="line.625"></a>
<FONT color="green">626</FONT>                    this.wbar = v;<a name="line.626"></a>
<FONT color="green">627</FONT>                }<a name="line.627"></a>
<FONT color="green">628</FONT>                updateNorms();<a name="line.628"></a>
<FONT color="green">629</FONT>            }<a name="line.629"></a>
<FONT color="green">630</FONT>    <a name="line.630"></a>
<FONT color="green">631</FONT>            /**<a name="line.631"></a>
<FONT color="green">632</FONT>             * Performs the next iteration of the algorithm. The iteration count<a name="line.632"></a>
<FONT color="green">633</FONT>             * should be incremented prior to calling this method. On return, the<a name="line.633"></a>
<FONT color="green">634</FONT>             * value of the state variables of {@code this} object correspond to the<a name="line.634"></a>
<FONT color="green">635</FONT>             * current iteration count {@code k}.<a name="line.635"></a>
<FONT color="green">636</FONT>             */<a name="line.636"></a>
<FONT color="green">637</FONT>            void update() {<a name="line.637"></a>
<FONT color="green">638</FONT>                final RealVector v = y.mapMultiply(1. / beta);<a name="line.638"></a>
<FONT color="green">639</FONT>                y = a.operate(v);<a name="line.639"></a>
<FONT color="green">640</FONT>                daxpbypz(-shift, v, -beta / oldb, r1, y);<a name="line.640"></a>
<FONT color="green">641</FONT>                final double alpha = v.dotProduct(y);<a name="line.641"></a>
<FONT color="green">642</FONT>                /*<a name="line.642"></a>
<FONT color="green">643</FONT>                 * At this point<a name="line.643"></a>
<FONT color="green">644</FONT>                 *   v     = P' * v[k],<a name="line.644"></a>
<FONT color="green">645</FONT>                 *   y     = (A - shift * I) * P' * v[k] - beta[k] * M^(-1) * P' * v[k-1],<a name="line.645"></a>
<FONT color="green">646</FONT>                 *   alpha = v'[k] * P * (A - shift * I) * P' * v[k]<a name="line.646"></a>
<FONT color="green">647</FONT>                 *           - beta[k] * v[k]' * P * M^(-1) * P' * v[k-1]<a name="line.647"></a>
<FONT color="green">648</FONT>                 *         = v'[k] * P * (A - shift * I) * P' * v[k]<a name="line.648"></a>
<FONT color="green">649</FONT>                 *           - beta[k] * v[k]' * v[k-1]<a name="line.649"></a>
<FONT color="green">650</FONT>                 *         = alpha[k].<a name="line.650"></a>
<FONT color="green">651</FONT>                 */<a name="line.651"></a>
<FONT color="green">652</FONT>                daxpy(-alpha / beta, r2, y);<a name="line.652"></a>
<FONT color="green">653</FONT>                /*<a name="line.653"></a>
<FONT color="green">654</FONT>                 * At this point<a name="line.654"></a>
<FONT color="green">655</FONT>                 *   y = (A - shift * I) * P' * v[k] - alpha[k] * M^(-1) * P' * v[k]<a name="line.655"></a>
<FONT color="green">656</FONT>                 *       - beta[k] * M^(-1) * P' * v[k-1]<a name="line.656"></a>
<FONT color="green">657</FONT>                 *     = M^(-1) * P' * (P * (A - shift * I) * P' * v[k] -alpha[k] * v[k]<a name="line.657"></a>
<FONT color="green">658</FONT>                 *       - beta[k] * v[k-1])<a name="line.658"></a>
<FONT color="green">659</FONT>                 *     = beta[k+1] * M^(-1) * P' * v[k+1],<a name="line.659"></a>
<FONT color="green">660</FONT>                 * from Paige and Saunders (1975), equation (3.2).<a name="line.660"></a>
<FONT color="green">661</FONT>                 *<a name="line.661"></a>
<FONT color="green">662</FONT>                 * WATCH-IT: the two following lines work only because y is no longer<a name="line.662"></a>
<FONT color="green">663</FONT>                 * updated up to the end of the present iteration, and is<a name="line.663"></a>
<FONT color="green">664</FONT>                 * reinitialized at the beginning of the next iteration.<a name="line.664"></a>
<FONT color="green">665</FONT>                 */<a name="line.665"></a>
<FONT color="green">666</FONT>                r1 = r2;<a name="line.666"></a>
<FONT color="green">667</FONT>                r2 = y;<a name="line.667"></a>
<FONT color="green">668</FONT>                if (m != null) {<a name="line.668"></a>
<FONT color="green">669</FONT>                    y = m.operate(r2);<a name="line.669"></a>
<FONT color="green">670</FONT>                }<a name="line.670"></a>
<FONT color="green">671</FONT>                oldb = beta;<a name="line.671"></a>
<FONT color="green">672</FONT>                beta = r2.dotProduct(y);<a name="line.672"></a>
<FONT color="green">673</FONT>                if (beta &lt; 0.) {<a name="line.673"></a>
<FONT color="green">674</FONT>                    throwNPDLOException(m, y);<a name="line.674"></a>
<FONT color="green">675</FONT>                }<a name="line.675"></a>
<FONT color="green">676</FONT>                beta = FastMath.sqrt(beta);<a name="line.676"></a>
<FONT color="green">677</FONT>                /*<a name="line.677"></a>
<FONT color="green">678</FONT>                 * At this point<a name="line.678"></a>
<FONT color="green">679</FONT>                 *   r1 = beta[k] * M^(-1) * P' * v[k],<a name="line.679"></a>
<FONT color="green">680</FONT>                 *   r2 = beta[k+1] * M^(-1) * P' * v[k+1],<a name="line.680"></a>
<FONT color="green">681</FONT>                 *   y  = beta[k+1] * P' * v[k+1],<a name="line.681"></a>
<FONT color="green">682</FONT>                 *   oldb = beta[k],<a name="line.682"></a>
<FONT color="green">683</FONT>                 *   beta = beta[k+1].<a name="line.683"></a>
<FONT color="green">684</FONT>                 */<a name="line.684"></a>
<FONT color="green">685</FONT>                tnorm += alpha * alpha + oldb * oldb + beta * beta;<a name="line.685"></a>
<FONT color="green">686</FONT>                /*<a name="line.686"></a>
<FONT color="green">687</FONT>                 * Compute the next plane rotation for Q. See Paige and Saunders<a name="line.687"></a>
<FONT color="green">688</FONT>                 * (1975), equation (5.6), with<a name="line.688"></a>
<FONT color="green">689</FONT>                 *   gamma = gamma[k-1],<a name="line.689"></a>
<FONT color="green">690</FONT>                 *   c     = c[k-1],<a name="line.690"></a>
<FONT color="green">691</FONT>                 *   s     = s[k-1].<a name="line.691"></a>
<FONT color="green">692</FONT>                 */<a name="line.692"></a>
<FONT color="green">693</FONT>                final double gamma = FastMath.sqrt(gbar * gbar + oldb * oldb);<a name="line.693"></a>
<FONT color="green">694</FONT>                final double c = gbar / gamma;<a name="line.694"></a>
<FONT color="green">695</FONT>                final double s = oldb / gamma;<a name="line.695"></a>
<FONT color="green">696</FONT>                /*<a name="line.696"></a>
<FONT color="green">697</FONT>                 * The relations<a name="line.697"></a>
<FONT color="green">698</FONT>                 *   gbar[k] = s[k-1] * (-c[k-2] * beta[k]) - c[k-1] * alpha[k]<a name="line.698"></a>
<FONT color="green">699</FONT>                 *           = s[k-1] * dbar[k] - c[k-1] * alpha[k],<a name="line.699"></a>
<FONT color="green">700</FONT>                 *   delta[k] = c[k-1] * dbar[k] + s[k-1] * alpha[k],<a name="line.700"></a>
<FONT color="green">701</FONT>                 * are not stated in Paige and Saunders (1975), but can be retrieved<a name="line.701"></a>
<FONT color="green">702</FONT>                 * by expanding the (k, k-1) and (k, k) coefficients of the matrix in<a name="line.702"></a>
<FONT color="green">703</FONT>                 * equation (5.5).<a name="line.703"></a>
<FONT color="green">704</FONT>                 */<a name="line.704"></a>
<FONT color="green">705</FONT>                final double deltak = c * dbar + s * alpha;<a name="line.705"></a>
<FONT color="green">706</FONT>                gbar = s * dbar - c * alpha;<a name="line.706"></a>
<FONT color="green">707</FONT>                final double eps = s * beta;<a name="line.707"></a>
<FONT color="green">708</FONT>                dbar = -c * beta;<a name="line.708"></a>
<FONT color="green">709</FONT>                final double zeta = gammaZeta / gamma;<a name="line.709"></a>
<FONT color="green">710</FONT>                /*<a name="line.710"></a>
<FONT color="green">711</FONT>                 * At this point<a name="line.711"></a>
<FONT color="green">712</FONT>                 *   gbar   = gbar[k]<a name="line.712"></a>
<FONT color="green">713</FONT>                 *   deltak = delta[k]<a name="line.713"></a>
<FONT color="green">714</FONT>                 *   eps    = eps[k+1]<a name="line.714"></a>
<FONT color="green">715</FONT>                 *   dbar   = dbar[k+1]<a name="line.715"></a>
<FONT color="green">716</FONT>                 *   zeta   = zeta[k-1]<a name="line.716"></a>
<FONT color="green">717</FONT>                 */<a name="line.717"></a>
<FONT color="green">718</FONT>                final double zetaC = zeta * c;<a name="line.718"></a>
<FONT color="green">719</FONT>                final double zetaS = zeta * s;<a name="line.719"></a>
<FONT color="green">720</FONT>                final int n = xL.getDimension();<a name="line.720"></a>
<FONT color="green">721</FONT>                for (int i = 0; i &lt; n; i++) {<a name="line.721"></a>
<FONT color="green">722</FONT>                    final double xi = xL.getEntry(i);<a name="line.722"></a>
<FONT color="green">723</FONT>                    final double vi = v.getEntry(i);<a name="line.723"></a>
<FONT color="green">724</FONT>                    final double wi = wbar.getEntry(i);<a name="line.724"></a>
<FONT color="green">725</FONT>                    xL.setEntry(i, xi + wi * zetaC + vi * zetaS);<a name="line.725"></a>
<FONT color="green">726</FONT>                    wbar.setEntry(i, wi * s - vi * c);<a name="line.726"></a>
<FONT color="green">727</FONT>                }<a name="line.727"></a>
<FONT color="green">728</FONT>                /*<a name="line.728"></a>
<FONT color="green">729</FONT>                 * At this point<a name="line.729"></a>
<FONT color="green">730</FONT>                 *   x = xL[k-1],<a name="line.730"></a>
<FONT color="green">731</FONT>                 *   ptwbar = P' wbar[k],<a name="line.731"></a>
<FONT color="green">732</FONT>                 * see Paige and Saunders (1975), equations (5.9) and (5.10).<a name="line.732"></a>
<FONT color="green">733</FONT>                 */<a name="line.733"></a>
<FONT color="green">734</FONT>                bstep += snprod * c * zeta;<a name="line.734"></a>
<FONT color="green">735</FONT>                snprod *= s;<a name="line.735"></a>
<FONT color="green">736</FONT>                gmax = FastMath.max(gmax, gamma);<a name="line.736"></a>
<FONT color="green">737</FONT>                gmin = FastMath.min(gmin, gamma);<a name="line.737"></a>
<FONT color="green">738</FONT>                ynorm2 += zeta * zeta;<a name="line.738"></a>
<FONT color="green">739</FONT>                gammaZeta = minusEpsZeta - deltak * zeta;<a name="line.739"></a>
<FONT color="green">740</FONT>                minusEpsZeta = -eps * zeta;<a name="line.740"></a>
<FONT color="green">741</FONT>                /*<a name="line.741"></a>
<FONT color="green">742</FONT>                 * At this point<a name="line.742"></a>
<FONT color="green">743</FONT>                 *   snprod       = s[1] * ... * s[k-1],<a name="line.743"></a>
<FONT color="green">744</FONT>                 *   gmax         = max(|alpha[1]|, gamma[1], ..., gamma[k-1]),<a name="line.744"></a>
<FONT color="green">745</FONT>                 *   gmin         = min(|alpha[1]|, gamma[1], ..., gamma[k-1]),<a name="line.745"></a>
<FONT color="green">746</FONT>                 *   ynorm2       = zeta[1]^2 + ... + zeta[k-1]^2,<a name="line.746"></a>
<FONT color="green">747</FONT>                 *   gammaZeta    = gamma[k] * zeta[k],<a name="line.747"></a>
<FONT color="green">748</FONT>                 *   minusEpsZeta = -eps[k+1] * zeta[k-1].<a name="line.748"></a>
<FONT color="green">749</FONT>                 * The relation for gammaZeta can be retrieved from Paige and<a name="line.749"></a>
<FONT color="green">750</FONT>                 * Saunders (1975), equation (5.4a), last line of the vector<a name="line.750"></a>
<FONT color="green">751</FONT>                 * gbar[k] * zbar[k] = -eps[k] * zeta[k-2] - delta[k] * zeta[k-1].<a name="line.751"></a>
<FONT color="green">752</FONT>                 */<a name="line.752"></a>
<FONT color="green">753</FONT>                updateNorms();<a name="line.753"></a>
<FONT color="green">754</FONT>            }<a name="line.754"></a>
<FONT color="green">755</FONT>    <a name="line.755"></a>
<FONT color="green">756</FONT>            /**<a name="line.756"></a>
<FONT color="green">757</FONT>             * Computes the norms of the residuals, and checks for convergence.<a name="line.757"></a>
<FONT color="green">758</FONT>             * Updates {@link #lqnorm} and {@link #cgnorm}.<a name="line.758"></a>
<FONT color="green">759</FONT>             */<a name="line.759"></a>
<FONT color="green">760</FONT>            private void updateNorms() {<a name="line.760"></a>
<FONT color="green">761</FONT>                final double anorm = FastMath.sqrt(tnorm);<a name="line.761"></a>
<FONT color="green">762</FONT>                final double ynorm = FastMath.sqrt(ynorm2);<a name="line.762"></a>
<FONT color="green">763</FONT>                final double epsa = anorm * MACH_PREC;<a name="line.763"></a>
<FONT color="green">764</FONT>                final double epsx = anorm * ynorm * MACH_PREC;<a name="line.764"></a>
<FONT color="green">765</FONT>                final double epsr = anorm * ynorm * delta;<a name="line.765"></a>
<FONT color="green">766</FONT>                final double diag = gbar == 0. ? epsa : gbar;<a name="line.766"></a>
<FONT color="green">767</FONT>                lqnorm = FastMath.sqrt(gammaZeta * gammaZeta +<a name="line.767"></a>
<FONT color="green">768</FONT>                                       minusEpsZeta * minusEpsZeta);<a name="line.768"></a>
<FONT color="green">769</FONT>                final double qrnorm = snprod * beta1;<a name="line.769"></a>
<FONT color="green">770</FONT>                cgnorm = qrnorm * beta / FastMath.abs(diag);<a name="line.770"></a>
<FONT color="green">771</FONT>    <a name="line.771"></a>
<FONT color="green">772</FONT>                /*<a name="line.772"></a>
<FONT color="green">773</FONT>                 * Estimate cond(A). In this version we look at the diagonals of L<a name="line.773"></a>
<FONT color="green">774</FONT>                 * in the factorization of the tridiagonal matrix, T = L * Q.<a name="line.774"></a>
<FONT color="green">775</FONT>                 * Sometimes, T[k] can be misleadingly ill-conditioned when T[k+1]<a name="line.775"></a>
<FONT color="green">776</FONT>                 * is not, so we must be careful not to overestimate acond.<a name="line.776"></a>
<FONT color="green">777</FONT>                 */<a name="line.777"></a>
<FONT color="green">778</FONT>                final double acond;<a name="line.778"></a>
<FONT color="green">779</FONT>                if (lqnorm &lt;= cgnorm) {<a name="line.779"></a>
<FONT color="green">780</FONT>                    acond = gmax / gmin;<a name="line.780"></a>
<FONT color="green">781</FONT>                } else {<a name="line.781"></a>
<FONT color="green">782</FONT>                    acond = gmax / FastMath.min(gmin, FastMath.abs(diag));<a name="line.782"></a>
<FONT color="green">783</FONT>                }<a name="line.783"></a>
<FONT color="green">784</FONT>                if (acond * MACH_PREC &gt;= 0.1) {<a name="line.784"></a>
<FONT color="green">785</FONT>                    throw new IllConditionedOperatorException(acond);<a name="line.785"></a>
<FONT color="green">786</FONT>                }<a name="line.786"></a>
<FONT color="green">787</FONT>                if (beta1 &lt;= epsx) {<a name="line.787"></a>
<FONT color="green">788</FONT>                    /*<a name="line.788"></a>
<FONT color="green">789</FONT>                     * x has converged to an eigenvector of A corresponding to the<a name="line.789"></a>
<FONT color="green">790</FONT>                     * eigenvalue shift.<a name="line.790"></a>
<FONT color="green">791</FONT>                     */<a name="line.791"></a>
<FONT color="green">792</FONT>                    throw new SingularOperatorException();<a name="line.792"></a>
<FONT color="green">793</FONT>                }<a name="line.793"></a>
<FONT color="green">794</FONT>                rnorm = FastMath.min(cgnorm, lqnorm);<a name="line.794"></a>
<FONT color="green">795</FONT>                hasConverged = (cgnorm &lt;= epsx) || (cgnorm &lt;= epsr);<a name="line.795"></a>
<FONT color="green">796</FONT>            }<a name="line.796"></a>
<FONT color="green">797</FONT>    <a name="line.797"></a>
<FONT color="green">798</FONT>            /**<a name="line.798"></a>
<FONT color="green">799</FONT>             * Returns {@code true} if the default stopping criterion is fulfilled.<a name="line.799"></a>
<FONT color="green">800</FONT>             *<a name="line.800"></a>
<FONT color="green">801</FONT>             * @return {@code true} if convergence of the iterations has occured<a name="line.801"></a>
<FONT color="green">802</FONT>             */<a name="line.802"></a>
<FONT color="green">803</FONT>            boolean hasConverged() {<a name="line.803"></a>
<FONT color="green">804</FONT>                return hasConverged;<a name="line.804"></a>
<FONT color="green">805</FONT>            }<a name="line.805"></a>
<FONT color="green">806</FONT>    <a name="line.806"></a>
<FONT color="green">807</FONT>            /**<a name="line.807"></a>
<FONT color="green">808</FONT>             * Returns {@code true} if the right-hand side vector is zero exactly.<a name="line.808"></a>
<FONT color="green">809</FONT>             *<a name="line.809"></a>
<FONT color="green">810</FONT>             * @return the boolean value of {@code b == 0}<a name="line.810"></a>
<FONT color="green">811</FONT>             */<a name="line.811"></a>
<FONT color="green">812</FONT>            boolean bEqualsNullVector() {<a name="line.812"></a>
<FONT color="green">813</FONT>                return bIsNull;<a name="line.813"></a>
<FONT color="green">814</FONT>            }<a name="line.814"></a>
<FONT color="green">815</FONT>    <a name="line.815"></a>
<FONT color="green">816</FONT>            /**<a name="line.816"></a>
<FONT color="green">817</FONT>             * Returns {@code true} if {@code beta} is essentially zero. This method<a name="line.817"></a>
<FONT color="green">818</FONT>             * is used to check for early stop of the iterations.<a name="line.818"></a>
<FONT color="green">819</FONT>             *<a name="line.819"></a>
<FONT color="green">820</FONT>             * @return {@code true} if {@code beta &lt; }{@link #MACH_PREC}<a name="line.820"></a>
<FONT color="green">821</FONT>             */<a name="line.821"></a>
<FONT color="green">822</FONT>            boolean betaEqualsZero() {<a name="line.822"></a>
<FONT color="green">823</FONT>                return beta &lt; MACH_PREC;<a name="line.823"></a>
<FONT color="green">824</FONT>            }<a name="line.824"></a>
<FONT color="green">825</FONT>    <a name="line.825"></a>
<FONT color="green">826</FONT>            /**<a name="line.826"></a>
<FONT color="green">827</FONT>             * Returns the norm of the updated, preconditioned residual.<a name="line.827"></a>
<FONT color="green">828</FONT>             *<a name="line.828"></a>
<FONT color="green">829</FONT>             * @return the norm of the residual, ||P * r||<a name="line.829"></a>
<FONT color="green">830</FONT>             */<a name="line.830"></a>
<FONT color="green">831</FONT>            double getNormOfResidual() {<a name="line.831"></a>
<FONT color="green">832</FONT>                return rnorm;<a name="line.832"></a>
<FONT color="green">833</FONT>            }<a name="line.833"></a>
<FONT color="green">834</FONT>        }<a name="line.834"></a>
<FONT color="green">835</FONT>    <a name="line.835"></a>
<FONT color="green">836</FONT>        /** Key for the exception context. */<a name="line.836"></a>
<FONT color="green">837</FONT>        private static final String OPERATOR = "operator";<a name="line.837"></a>
<FONT color="green">838</FONT>    <a name="line.838"></a>
<FONT color="green">839</FONT>        /** Key for the exception context. */<a name="line.839"></a>
<FONT color="green">840</FONT>        private static final String THRESHOLD = "threshold";<a name="line.840"></a>
<FONT color="green">841</FONT>    <a name="line.841"></a>
<FONT color="green">842</FONT>        /** Key for the exception context. */<a name="line.842"></a>
<FONT color="green">843</FONT>        private static final String VECTOR = "vector";<a name="line.843"></a>
<FONT color="green">844</FONT>    <a name="line.844"></a>
<FONT color="green">845</FONT>        /** Key for the exception context. */<a name="line.845"></a>
<FONT color="green">846</FONT>        private static final String VECTOR1 = "vector1";<a name="line.846"></a>
<FONT color="green">847</FONT>    <a name="line.847"></a>
<FONT color="green">848</FONT>        /** Key for the exception context. */<a name="line.848"></a>
<FONT color="green">849</FONT>        private static final String VECTOR2 = "vector2";<a name="line.849"></a>
<FONT color="green">850</FONT>    <a name="line.850"></a>
<FONT color="green">851</FONT>        /** {@code true} if symmetry of matrix and conditioner must be checked. */<a name="line.851"></a>
<FONT color="green">852</FONT>        private final boolean check;<a name="line.852"></a>
<FONT color="green">853</FONT>    <a name="line.853"></a>
<FONT color="green">854</FONT>        /**<a name="line.854"></a>
<FONT color="green">855</FONT>         * The value of the custom tolerance &amp;delta; for the default stopping<a name="line.855"></a>
<FONT color="green">856</FONT>         * criterion.<a name="line.856"></a>
<FONT color="green">857</FONT>         */<a name="line.857"></a>
<FONT color="green">858</FONT>        private final double delta;<a name="line.858"></a>
<FONT color="green">859</FONT>    <a name="line.859"></a>
<FONT color="green">860</FONT>        /**<a name="line.860"></a>
<FONT color="green">861</FONT>         * Creates a new instance of this class, with &lt;a href="#stopcrit"&gt;default<a name="line.861"></a>
<FONT color="green">862</FONT>         * stopping criterion&lt;/a&gt;. Note that setting {@code check} to {@code true}<a name="line.862"></a>
<FONT color="green">863</FONT>         * entails an extra matrix-vector product in the initial phase.<a name="line.863"></a>
<FONT color="green">864</FONT>         *<a name="line.864"></a>
<FONT color="green">865</FONT>         * @param maxIterations the maximum number of iterations<a name="line.865"></a>
<FONT color="green">866</FONT>         * @param delta the &amp;delta; parameter for the default stopping criterion<a name="line.866"></a>
<FONT color="green">867</FONT>         * @param check {@code true} if self-adjointedness of both matrix and<a name="line.867"></a>
<FONT color="green">868</FONT>         * preconditioner should be checked<a name="line.868"></a>
<FONT color="green">869</FONT>         */<a name="line.869"></a>
<FONT color="green">870</FONT>        public SymmLQ(final int maxIterations, final double delta,<a name="line.870"></a>
<FONT color="green">871</FONT>                      final boolean check) {<a name="line.871"></a>
<FONT color="green">872</FONT>            super(maxIterations);<a name="line.872"></a>
<FONT color="green">873</FONT>            this.delta = delta;<a name="line.873"></a>
<FONT color="green">874</FONT>            this.check = check;<a name="line.874"></a>
<FONT color="green">875</FONT>        }<a name="line.875"></a>
<FONT color="green">876</FONT>    <a name="line.876"></a>
<FONT color="green">877</FONT>        /**<a name="line.877"></a>
<FONT color="green">878</FONT>         * Creates a new instance of this class, with &lt;a href="#stopcrit"&gt;default<a name="line.878"></a>
<FONT color="green">879</FONT>         * stopping criterion&lt;/a&gt; and custom iteration manager. Note that setting<a name="line.879"></a>
<FONT color="green">880</FONT>         * {@code check} to {@code true} entails an extra matrix-vector product in<a name="line.880"></a>
<FONT color="green">881</FONT>         * the initial phase.<a name="line.881"></a>
<FONT color="green">882</FONT>         *<a name="line.882"></a>
<FONT color="green">883</FONT>         * @param manager the custom iteration manager<a name="line.883"></a>
<FONT color="green">884</FONT>         * @param delta the &amp;delta; parameter for the default stopping criterion<a name="line.884"></a>
<FONT color="green">885</FONT>         * @param check {@code true} if self-adjointedness of both matrix and<a name="line.885"></a>
<FONT color="green">886</FONT>         * preconditioner should be checked<a name="line.886"></a>
<FONT color="green">887</FONT>         */<a name="line.887"></a>
<FONT color="green">888</FONT>        public SymmLQ(final IterationManager manager, final double delta,<a name="line.888"></a>
<FONT color="green">889</FONT>                      final boolean check) {<a name="line.889"></a>
<FONT color="green">890</FONT>            super(manager);<a name="line.890"></a>
<FONT color="green">891</FONT>            this.delta = delta;<a name="line.891"></a>
<FONT color="green">892</FONT>            this.check = check;<a name="line.892"></a>
<FONT color="green">893</FONT>        }<a name="line.893"></a>
<FONT color="green">894</FONT>    <a name="line.894"></a>
<FONT color="green">895</FONT>        /**<a name="line.895"></a>
<FONT color="green">896</FONT>         * Returns {@code true} if symmetry of the matrix, and symmetry as well as<a name="line.896"></a>
<FONT color="green">897</FONT>         * positive definiteness of the preconditioner should be checked.<a name="line.897"></a>
<FONT color="green">898</FONT>         *<a name="line.898"></a>
<FONT color="green">899</FONT>         * @return {@code true} if the tests are to be performed<a name="line.899"></a>
<FONT color="green">900</FONT>         */<a name="line.900"></a>
<FONT color="green">901</FONT>        public final boolean getCheck() {<a name="line.901"></a>
<FONT color="green">902</FONT>            return check;<a name="line.902"></a>
<FONT color="green">903</FONT>        }<a name="line.903"></a>
<FONT color="green">904</FONT>    <a name="line.904"></a>
<FONT color="green">905</FONT>        /**<a name="line.905"></a>
<FONT color="green">906</FONT>         * {@inheritDoc}<a name="line.906"></a>
<FONT color="green">907</FONT>         *<a name="line.907"></a>
<FONT color="green">908</FONT>         * @throws NonSelfAdjointOperatorException if {@link #getCheck()} is<a name="line.908"></a>
<FONT color="green">909</FONT>         * {@code true}, and {@code a} or {@code m} is not self-adjoint<a name="line.909"></a>
<FONT color="green">910</FONT>         * @throws NonPositiveDefiniteOperatorException if {@code m} is not<a name="line.910"></a>
<FONT color="green">911</FONT>         * positive definite<a name="line.911"></a>
<FONT color="green">912</FONT>         * @throws IllConditionedOperatorException if {@code a} is ill-conditioned<a name="line.912"></a>
<FONT color="green">913</FONT>         */<a name="line.913"></a>
<FONT color="green">914</FONT>        @Override<a name="line.914"></a>
<FONT color="green">915</FONT>        public RealVector solve(final RealLinearOperator a,<a name="line.915"></a>
<FONT color="green">916</FONT>            final RealLinearOperator m, final RealVector b) throws<a name="line.916"></a>
<FONT color="green">917</FONT>            NullArgumentException, NonSquareOperatorException,<a name="line.917"></a>
<FONT color="green">918</FONT>            DimensionMismatchException, MaxCountExceededException,<a name="line.918"></a>
<FONT color="green">919</FONT>            NonSelfAdjointOperatorException, NonPositiveDefiniteOperatorException,<a name="line.919"></a>
<FONT color="green">920</FONT>            IllConditionedOperatorException {<a name="line.920"></a>
<FONT color="green">921</FONT>            MathUtils.checkNotNull(a);<a name="line.921"></a>
<FONT color="green">922</FONT>            final RealVector x = new ArrayRealVector(a.getColumnDimension());<a name="line.922"></a>
<FONT color="green">923</FONT>            return solveInPlace(a, m, b, x, false, 0.);<a name="line.923"></a>
<FONT color="green">924</FONT>        }<a name="line.924"></a>
<FONT color="green">925</FONT>    <a name="line.925"></a>
<FONT color="green">926</FONT>        /**<a name="line.926"></a>
<FONT color="green">927</FONT>         * Returns an estimate of the solution to the linear system (A - shift<a name="line.927"></a>
<FONT color="green">928</FONT>         * &amp;middot; I) &amp;middot; x = b.<a name="line.928"></a>
<FONT color="green">929</FONT>         * &lt;p&gt;<a name="line.929"></a>
<FONT color="green">930</FONT>         * If the solution x is expected to contain a large multiple of {@code b}<a name="line.930"></a>
<FONT color="green">931</FONT>         * (as in Rayleigh-quotient iteration), then better precision may be<a name="line.931"></a>
<FONT color="green">932</FONT>         * achieved with {@code goodb} set to {@code true}; this however requires an<a name="line.932"></a>
<FONT color="green">933</FONT>         * extra call to the preconditioner.<a name="line.933"></a>
<FONT color="green">934</FONT>         * &lt;/p&gt;<a name="line.934"></a>
<FONT color="green">935</FONT>         * &lt;p&gt;<a name="line.935"></a>
<FONT color="green">936</FONT>         * {@code shift} should be zero if the system A &amp;middot; x = b is to be<a name="line.936"></a>
<FONT color="green">937</FONT>         * solved. Otherwise, it could be an approximation to an eigenvalue of A,<a name="line.937"></a>
<FONT color="green">938</FONT>         * such as the Rayleigh quotient b&lt;sup&gt;T&lt;/sup&gt; &amp;middot; A &amp;middot; b /<a name="line.938"></a>
<FONT color="green">939</FONT>         * (b&lt;sup&gt;T&lt;/sup&gt; &amp;middot; b) corresponding to the vector b. If b is<a name="line.939"></a>
<FONT color="green">940</FONT>         * sufficiently like an eigenvector corresponding to an eigenvalue near<a name="line.940"></a>
<FONT color="green">941</FONT>         * shift, then the computed x may have very large components. When<a name="line.941"></a>
<FONT color="green">942</FONT>         * normalized, x may be closer to an eigenvector than b.<a name="line.942"></a>
<FONT color="green">943</FONT>         * &lt;/p&gt;<a name="line.943"></a>
<FONT color="green">944</FONT>         *<a name="line.944"></a>
<FONT color="green">945</FONT>         * @param a the linear operator A of the system<a name="line.945"></a>
<FONT color="green">946</FONT>         * @param m the preconditioner, M (can be {@code null})<a name="line.946"></a>
<FONT color="green">947</FONT>         * @param b the right-hand side vector<a name="line.947"></a>
<FONT color="green">948</FONT>         * @param goodb usually {@code false}, except if {@code x} is expected to<a name="line.948"></a>
<FONT color="green">949</FONT>         * contain a large multiple of {@code b}<a name="line.949"></a>
<FONT color="green">950</FONT>         * @param shift the amount to be subtracted to all diagonal elements of A<a name="line.950"></a>
<FONT color="green">951</FONT>         * @return a reference to {@code x} (shallow copy)<a name="line.951"></a>
<FONT color="green">952</FONT>         * @throws NullArgumentException if one of the parameters is {@code null}<a name="line.952"></a>
<FONT color="green">953</FONT>         * @throws NonSquareOperatorException if {@code a} or {@code m} is not square<a name="line.953"></a>
<FONT color="green">954</FONT>         * @throws DimensionMismatchException if {@code m} or {@code b} have dimensions<a name="line.954"></a>
<FONT color="green">955</FONT>         * inconsistent with {@code a}<a name="line.955"></a>
<FONT color="green">956</FONT>         * @throws MaxCountExceededException at exhaustion of the iteration count,<a name="line.956"></a>
<FONT color="green">957</FONT>         * unless a custom<a name="line.957"></a>
<FONT color="green">958</FONT>         * {@link org.apache.commons.math3.util.Incrementor.MaxCountExceededCallback callback}<a name="line.958"></a>
<FONT color="green">959</FONT>         * has been set at construction of the {@link IterationManager}<a name="line.959"></a>
<FONT color="green">960</FONT>         * @throws NonSelfAdjointOperatorException if {@link #getCheck()} is<a name="line.960"></a>
<FONT color="green">961</FONT>         * {@code true}, and {@code a} or {@code m} is not self-adjoint<a name="line.961"></a>
<FONT color="green">962</FONT>         * @throws NonPositiveDefiniteOperatorException if {@code m} is not<a name="line.962"></a>
<FONT color="green">963</FONT>         * positive definite<a name="line.963"></a>
<FONT color="green">964</FONT>         * @throws IllConditionedOperatorException if {@code a} is ill-conditioned<a name="line.964"></a>
<FONT color="green">965</FONT>         */<a name="line.965"></a>
<FONT color="green">966</FONT>        public RealVector solve(final RealLinearOperator a,<a name="line.966"></a>
<FONT color="green">967</FONT>            final RealLinearOperator m, final RealVector b, final boolean goodb,<a name="line.967"></a>
<FONT color="green">968</FONT>            final double shift) throws NullArgumentException,<a name="line.968"></a>
<FONT color="green">969</FONT>            NonSquareOperatorException, DimensionMismatchException,<a name="line.969"></a>
<FONT color="green">970</FONT>            MaxCountExceededException, NonSelfAdjointOperatorException,<a name="line.970"></a>
<FONT color="green">971</FONT>            NonPositiveDefiniteOperatorException, IllConditionedOperatorException {<a name="line.971"></a>
<FONT color="green">972</FONT>            MathUtils.checkNotNull(a);<a name="line.972"></a>
<FONT color="green">973</FONT>            final RealVector x = new ArrayRealVector(a.getColumnDimension());<a name="line.973"></a>
<FONT color="green">974</FONT>            return solveInPlace(a, m, b, x, goodb, shift);<a name="line.974"></a>
<FONT color="green">975</FONT>        }<a name="line.975"></a>
<FONT color="green">976</FONT>    <a name="line.976"></a>
<FONT color="green">977</FONT>        /**<a name="line.977"></a>
<FONT color="green">978</FONT>         * {@inheritDoc}<a name="line.978"></a>
<FONT color="green">979</FONT>         *<a name="line.979"></a>
<FONT color="green">980</FONT>         * @param x not meaningful in this implementation; should not be considered<a name="line.980"></a>
<FONT color="green">981</FONT>         * as an initial guess (&lt;a href="#initguess"&gt;more&lt;/a&gt;)<a name="line.981"></a>
<FONT color="green">982</FONT>         * @throws NonSelfAdjointOperatorException if {@link #getCheck()} is<a name="line.982"></a>
<FONT color="green">983</FONT>         * {@code true}, and {@code a} or {@code m} is not self-adjoint<a name="line.983"></a>
<FONT color="green">984</FONT>         * @throws NonPositiveDefiniteOperatorException if {@code m} is not positive<a name="line.984"></a>
<FONT color="green">985</FONT>         * definite<a name="line.985"></a>
<FONT color="green">986</FONT>         * @throws IllConditionedOperatorException if {@code a} is ill-conditioned<a name="line.986"></a>
<FONT color="green">987</FONT>         */<a name="line.987"></a>
<FONT color="green">988</FONT>        @Override<a name="line.988"></a>
<FONT color="green">989</FONT>        public RealVector solve(final RealLinearOperator a,<a name="line.989"></a>
<FONT color="green">990</FONT>            final RealLinearOperator m, final RealVector b, final RealVector x)<a name="line.990"></a>
<FONT color="green">991</FONT>            throws NullArgumentException, NonSquareOperatorException,<a name="line.991"></a>
<FONT color="green">992</FONT>            DimensionMismatchException, NonSelfAdjointOperatorException,<a name="line.992"></a>
<FONT color="green">993</FONT>            NonPositiveDefiniteOperatorException, IllConditionedOperatorException,<a name="line.993"></a>
<FONT color="green">994</FONT>            MaxCountExceededException {<a name="line.994"></a>
<FONT color="green">995</FONT>            MathUtils.checkNotNull(x);<a name="line.995"></a>
<FONT color="green">996</FONT>            return solveInPlace(a, m, b, x.copy(), false, 0.);<a name="line.996"></a>
<FONT color="green">997</FONT>        }<a name="line.997"></a>
<FONT color="green">998</FONT>    <a name="line.998"></a>
<FONT color="green">999</FONT>        /**<a name="line.999"></a>
<FONT color="green">1000</FONT>         * {@inheritDoc}<a name="line.1000"></a>
<FONT color="green">1001</FONT>         *<a name="line.1001"></a>
<FONT color="green">1002</FONT>         * @throws NonSelfAdjointOperatorException if {@link #getCheck()} is<a name="line.1002"></a>
<FONT color="green">1003</FONT>         * {@code true}, and {@code a} is not self-adjoint<a name="line.1003"></a>
<FONT color="green">1004</FONT>         * @throws IllConditionedOperatorException if {@code a} is ill-conditioned<a name="line.1004"></a>
<FONT color="green">1005</FONT>         */<a name="line.1005"></a>
<FONT color="green">1006</FONT>        @Override<a name="line.1006"></a>
<FONT color="green">1007</FONT>        public RealVector solve(final RealLinearOperator a, final RealVector b)<a name="line.1007"></a>
<FONT color="green">1008</FONT>            throws NullArgumentException, NonSquareOperatorException,<a name="line.1008"></a>
<FONT color="green">1009</FONT>            DimensionMismatchException, NonSelfAdjointOperatorException,<a name="line.1009"></a>
<FONT color="green">1010</FONT>            IllConditionedOperatorException, MaxCountExceededException {<a name="line.1010"></a>
<FONT color="green">1011</FONT>            MathUtils.checkNotNull(a);<a name="line.1011"></a>
<FONT color="green">1012</FONT>            final RealVector x = new ArrayRealVector(a.getColumnDimension());<a name="line.1012"></a>
<FONT color="green">1013</FONT>            x.set(0.);<a name="line.1013"></a>
<FONT color="green">1014</FONT>            return solveInPlace(a, null, b, x, false, 0.);<a name="line.1014"></a>
<FONT color="green">1015</FONT>        }<a name="line.1015"></a>
<FONT color="green">1016</FONT>    <a name="line.1016"></a>
<FONT color="green">1017</FONT>        /**<a name="line.1017"></a>
<FONT color="green">1018</FONT>         * Returns the solution to the system (A - shift &amp;middot; I) &amp;middot; x = b.<a name="line.1018"></a>
<FONT color="green">1019</FONT>         * &lt;p&gt;<a name="line.1019"></a>
<FONT color="green">1020</FONT>         * If the solution x is expected to contain a large multiple of {@code b}<a name="line.1020"></a>
<FONT color="green">1021</FONT>         * (as in Rayleigh-quotient iteration), then better precision may be<a name="line.1021"></a>
<FONT color="green">1022</FONT>         * achieved with {@code goodb} set to {@code true}.<a name="line.1022"></a>
<FONT color="green">1023</FONT>         * &lt;/p&gt;<a name="line.1023"></a>
<FONT color="green">1024</FONT>         * &lt;p&gt;<a name="line.1024"></a>
<FONT color="green">1025</FONT>         * {@code shift} should be zero if the system A &amp;middot; x = b is to be<a name="line.1025"></a>
<FONT color="green">1026</FONT>         * solved. Otherwise, it could be an approximation to an eigenvalue of A,<a name="line.1026"></a>
<FONT color="green">1027</FONT>         * such as the Rayleigh quotient b&lt;sup&gt;T&lt;/sup&gt; &amp;middot; A &amp;middot; b /<a name="line.1027"></a>
<FONT color="green">1028</FONT>         * (b&lt;sup&gt;T&lt;/sup&gt; &amp;middot; b) corresponding to the vector b. If b is<a name="line.1028"></a>
<FONT color="green">1029</FONT>         * sufficiently like an eigenvector corresponding to an eigenvalue near<a name="line.1029"></a>
<FONT color="green">1030</FONT>         * shift, then the computed x may have very large components. When<a name="line.1030"></a>
<FONT color="green">1031</FONT>         * normalized, x may be closer to an eigenvector than b.<a name="line.1031"></a>
<FONT color="green">1032</FONT>         * &lt;/p&gt;<a name="line.1032"></a>
<FONT color="green">1033</FONT>         *<a name="line.1033"></a>
<FONT color="green">1034</FONT>         * @param a the linear operator A of the system<a name="line.1034"></a>
<FONT color="green">1035</FONT>         * @param b the right-hand side vector<a name="line.1035"></a>
<FONT color="green">1036</FONT>         * @param goodb usually {@code false}, except if {@code x} is expected to<a name="line.1036"></a>
<FONT color="green">1037</FONT>         * contain a large multiple of {@code b}<a name="line.1037"></a>
<FONT color="green">1038</FONT>         * @param shift the amount to be subtracted to all diagonal elements of A<a name="line.1038"></a>
<FONT color="green">1039</FONT>         * @return a reference to {@code x}<a name="line.1039"></a>
<FONT color="green">1040</FONT>         * @throws NullArgumentException if one of the parameters is {@code null}<a name="line.1040"></a>
<FONT color="green">1041</FONT>         * @throws NonSquareOperatorException if {@code a} is not square<a name="line.1041"></a>
<FONT color="green">1042</FONT>         * @throws DimensionMismatchException if {@code b} has dimensions<a name="line.1042"></a>
<FONT color="green">1043</FONT>         * inconsistent with {@code a}<a name="line.1043"></a>
<FONT color="green">1044</FONT>         * @throws MaxCountExceededException at exhaustion of the iteration count,<a name="line.1044"></a>
<FONT color="green">1045</FONT>         * unless a custom<a name="line.1045"></a>
<FONT color="green">1046</FONT>         * {@link org.apache.commons.math3.util.Incrementor.MaxCountExceededCallback callback}<a name="line.1046"></a>
<FONT color="green">1047</FONT>         * has been set at construction of the {@link IterationManager}<a name="line.1047"></a>
<FONT color="green">1048</FONT>         * @throws NonSelfAdjointOperatorException if {@link #getCheck()} is<a name="line.1048"></a>
<FONT color="green">1049</FONT>         * {@code true}, and {@code a} is not self-adjoint<a name="line.1049"></a>
<FONT color="green">1050</FONT>         * @throws IllConditionedOperatorException if {@code a} is ill-conditioned<a name="line.1050"></a>
<FONT color="green">1051</FONT>         */<a name="line.1051"></a>
<FONT color="green">1052</FONT>        public RealVector solve(final RealLinearOperator a, final RealVector b,<a name="line.1052"></a>
<FONT color="green">1053</FONT>            final boolean goodb, final double shift) throws NullArgumentException,<a name="line.1053"></a>
<FONT color="green">1054</FONT>            NonSquareOperatorException, DimensionMismatchException,<a name="line.1054"></a>
<FONT color="green">1055</FONT>            NonSelfAdjointOperatorException, IllConditionedOperatorException,<a name="line.1055"></a>
<FONT color="green">1056</FONT>            MaxCountExceededException {<a name="line.1056"></a>
<FONT color="green">1057</FONT>            MathUtils.checkNotNull(a);<a name="line.1057"></a>
<FONT color="green">1058</FONT>            final RealVector x = new ArrayRealVector(a.getColumnDimension());<a name="line.1058"></a>
<FONT color="green">1059</FONT>            return solveInPlace(a, null, b, x, goodb, shift);<a name="line.1059"></a>
<FONT color="green">1060</FONT>        }<a name="line.1060"></a>
<FONT color="green">1061</FONT>    <a name="line.1061"></a>
<FONT color="green">1062</FONT>        /**<a name="line.1062"></a>
<FONT color="green">1063</FONT>         * {@inheritDoc}<a name="line.1063"></a>
<FONT color="green">1064</FONT>         *<a name="line.1064"></a>
<FONT color="green">1065</FONT>         * @param x not meaningful in this implementation; should not be considered<a name="line.1065"></a>
<FONT color="green">1066</FONT>         * as an initial guess (&lt;a href="#initguess"&gt;more&lt;/a&gt;)<a name="line.1066"></a>
<FONT color="green">1067</FONT>         * @throws NonSelfAdjointOperatorException if {@link #getCheck()} is<a name="line.1067"></a>
<FONT color="green">1068</FONT>         * {@code true}, and {@code a} is not self-adjoint<a name="line.1068"></a>
<FONT color="green">1069</FONT>         * @throws IllConditionedOperatorException if {@code a} is ill-conditioned<a name="line.1069"></a>
<FONT color="green">1070</FONT>         */<a name="line.1070"></a>
<FONT color="green">1071</FONT>        @Override<a name="line.1071"></a>
<FONT color="green">1072</FONT>        public RealVector solve(final RealLinearOperator a, final RealVector b,<a name="line.1072"></a>
<FONT color="green">1073</FONT>            final RealVector x) throws NullArgumentException,<a name="line.1073"></a>
<FONT color="green">1074</FONT>            NonSquareOperatorException, DimensionMismatchException,<a name="line.1074"></a>
<FONT color="green">1075</FONT>            NonSelfAdjointOperatorException, IllConditionedOperatorException,<a name="line.1075"></a>
<FONT color="green">1076</FONT>            MaxCountExceededException {<a name="line.1076"></a>
<FONT color="green">1077</FONT>            MathUtils.checkNotNull(x);<a name="line.1077"></a>
<FONT color="green">1078</FONT>            return solveInPlace(a, null, b, x.copy(), false, 0.);<a name="line.1078"></a>
<FONT color="green">1079</FONT>        }<a name="line.1079"></a>
<FONT color="green">1080</FONT>    <a name="line.1080"></a>
<FONT color="green">1081</FONT>        /**<a name="line.1081"></a>
<FONT color="green">1082</FONT>         * {@inheritDoc}<a name="line.1082"></a>
<FONT color="green">1083</FONT>         *<a name="line.1083"></a>
<FONT color="green">1084</FONT>         * @param x the vector to be updated with the solution; {@code x} should<a name="line.1084"></a>
<FONT color="green">1085</FONT>         * not be considered as an initial guess (&lt;a href="#initguess"&gt;more&lt;/a&gt;)<a name="line.1085"></a>
<FONT color="green">1086</FONT>         * @throws NonSelfAdjointOperatorException if {@link #getCheck()} is<a name="line.1086"></a>
<FONT color="green">1087</FONT>         * {@code true}, and {@code a} or {@code m} is not self-adjoint<a name="line.1087"></a>
<FONT color="green">1088</FONT>         * @throws NonPositiveDefiniteOperatorException if {@code m} is not<a name="line.1088"></a>
<FONT color="green">1089</FONT>         * positive definite<a name="line.1089"></a>
<FONT color="green">1090</FONT>         * @throws IllConditionedOperatorException if {@code a} is ill-conditioned<a name="line.1090"></a>
<FONT color="green">1091</FONT>         */<a name="line.1091"></a>
<FONT color="green">1092</FONT>        @Override<a name="line.1092"></a>
<FONT color="green">1093</FONT>        public RealVector solveInPlace(final RealLinearOperator a,<a name="line.1093"></a>
<FONT color="green">1094</FONT>            final RealLinearOperator m, final RealVector b, final RealVector x)<a name="line.1094"></a>
<FONT color="green">1095</FONT>            throws NullArgumentException, NonSquareOperatorException,<a name="line.1095"></a>
<FONT color="green">1096</FONT>            DimensionMismatchException, NonSelfAdjointOperatorException,<a name="line.1096"></a>
<FONT color="green">1097</FONT>            NonPositiveDefiniteOperatorException, IllConditionedOperatorException,<a name="line.1097"></a>
<FONT color="green">1098</FONT>            MaxCountExceededException {<a name="line.1098"></a>
<FONT color="green">1099</FONT>            return solveInPlace(a, m, b, x, false, 0.);<a name="line.1099"></a>
<FONT color="green">1100</FONT>        }<a name="line.1100"></a>
<FONT color="green">1101</FONT>    <a name="line.1101"></a>
<FONT color="green">1102</FONT>        /**<a name="line.1102"></a>
<FONT color="green">1103</FONT>         * Returns an estimate of the solution to the linear system (A - shift<a name="line.1103"></a>
<FONT color="green">1104</FONT>         * &amp;middot; I) &amp;middot; x = b. The solution is computed in-place.<a name="line.1104"></a>
<FONT color="green">1105</FONT>         * &lt;p&gt;<a name="line.1105"></a>
<FONT color="green">1106</FONT>         * If the solution x is expected to contain a large multiple of {@code b}<a name="line.1106"></a>
<FONT color="green">1107</FONT>         * (as in Rayleigh-quotient iteration), then better precision may be<a name="line.1107"></a>
<FONT color="green">1108</FONT>         * achieved with {@code goodb} set to {@code true}; this however requires an<a name="line.1108"></a>
<FONT color="green">1109</FONT>         * extra call to the preconditioner.<a name="line.1109"></a>
<FONT color="green">1110</FONT>         * &lt;/p&gt;<a name="line.1110"></a>
<FONT color="green">1111</FONT>         * &lt;p&gt;<a name="line.1111"></a>
<FONT color="green">1112</FONT>         * {@code shift} should be zero if the system A &amp;middot; x = b is to be<a name="line.1112"></a>
<FONT color="green">1113</FONT>         * solved. Otherwise, it could be an approximation to an eigenvalue of A,<a name="line.1113"></a>
<FONT color="green">1114</FONT>         * such as the Rayleigh quotient b&lt;sup&gt;T&lt;/sup&gt; &amp;middot; A &amp;middot; b /<a name="line.1114"></a>
<FONT color="green">1115</FONT>         * (b&lt;sup&gt;T&lt;/sup&gt; &amp;middot; b) corresponding to the vector b. If b is<a name="line.1115"></a>
<FONT color="green">1116</FONT>         * sufficiently like an eigenvector corresponding to an eigenvalue near<a name="line.1116"></a>
<FONT color="green">1117</FONT>         * shift, then the computed x may have very large components. When<a name="line.1117"></a>
<FONT color="green">1118</FONT>         * normalized, x may be closer to an eigenvector than b.<a name="line.1118"></a>
<FONT color="green">1119</FONT>         * &lt;/p&gt;<a name="line.1119"></a>
<FONT color="green">1120</FONT>         *<a name="line.1120"></a>
<FONT color="green">1121</FONT>         * @param a the linear operator A of the system<a name="line.1121"></a>
<FONT color="green">1122</FONT>         * @param m the preconditioner, M (can be {@code null})<a name="line.1122"></a>
<FONT color="green">1123</FONT>         * @param b the right-hand side vector<a name="line.1123"></a>
<FONT color="green">1124</FONT>         * @param x the vector to be updated with the solution; {@code x} should<a name="line.1124"></a>
<FONT color="green">1125</FONT>         * not be considered as an initial guess (&lt;a href="#initguess"&gt;more&lt;/a&gt;)<a name="line.1125"></a>
<FONT color="green">1126</FONT>         * @param goodb usually {@code false}, except if {@code x} is expected to<a name="line.1126"></a>
<FONT color="green">1127</FONT>         * contain a large multiple of {@code b}<a name="line.1127"></a>
<FONT color="green">1128</FONT>         * @param shift the amount to be subtracted to all diagonal elements of A<a name="line.1128"></a>
<FONT color="green">1129</FONT>         * @return a reference to {@code x} (shallow copy).<a name="line.1129"></a>
<FONT color="green">1130</FONT>         * @throws NullArgumentException if one of the parameters is {@code null}<a name="line.1130"></a>
<FONT color="green">1131</FONT>         * @throws NonSquareOperatorException if {@code a} or {@code m} is not square<a name="line.1131"></a>
<FONT color="green">1132</FONT>         * @throws DimensionMismatchException if {@code m}, {@code b} or {@code x}<a name="line.1132"></a>
<FONT color="green">1133</FONT>         * have dimensions inconsistent with {@code a}.<a name="line.1133"></a>
<FONT color="green">1134</FONT>         * @throws MaxCountExceededException at exhaustion of the iteration count,<a name="line.1134"></a>
<FONT color="green">1135</FONT>         * unless a custom<a name="line.1135"></a>
<FONT color="green">1136</FONT>         * {@link org.apache.commons.math3.util.Incrementor.MaxCountExceededCallback callback}<a name="line.1136"></a>
<FONT color="green">1137</FONT>         * has been set at construction of the {@link IterationManager}<a name="line.1137"></a>
<FONT color="green">1138</FONT>         * @throws NonSelfAdjointOperatorException if {@link #getCheck()} is<a name="line.1138"></a>
<FONT color="green">1139</FONT>         * {@code true}, and {@code a} or {@code m} is not self-adjoint<a name="line.1139"></a>
<FONT color="green">1140</FONT>         * @throws NonPositiveDefiniteOperatorException if {@code m} is not positive<a name="line.1140"></a>
<FONT color="green">1141</FONT>         * definite<a name="line.1141"></a>
<FONT color="green">1142</FONT>         * @throws IllConditionedOperatorException if {@code a} is ill-conditioned<a name="line.1142"></a>
<FONT color="green">1143</FONT>         */<a name="line.1143"></a>
<FONT color="green">1144</FONT>        public RealVector solveInPlace(final RealLinearOperator a,<a name="line.1144"></a>
<FONT color="green">1145</FONT>            final RealLinearOperator m, final RealVector b,<a name="line.1145"></a>
<FONT color="green">1146</FONT>            final RealVector x, final boolean goodb, final double shift)<a name="line.1146"></a>
<FONT color="green">1147</FONT>            throws NullArgumentException, NonSquareOperatorException,<a name="line.1147"></a>
<FONT color="green">1148</FONT>            DimensionMismatchException, NonSelfAdjointOperatorException,<a name="line.1148"></a>
<FONT color="green">1149</FONT>            NonPositiveDefiniteOperatorException, IllConditionedOperatorException,<a name="line.1149"></a>
<FONT color="green">1150</FONT>            MaxCountExceededException {<a name="line.1150"></a>
<FONT color="green">1151</FONT>            checkParameters(a, m, b, x);<a name="line.1151"></a>
<FONT color="green">1152</FONT>    <a name="line.1152"></a>
<FONT color="green">1153</FONT>            final IterationManager manager = getIterationManager();<a name="line.1153"></a>
<FONT color="green">1154</FONT>            /* Initialization counts as an iteration. */<a name="line.1154"></a>
<FONT color="green">1155</FONT>            manager.resetIterationCount();<a name="line.1155"></a>
<FONT color="green">1156</FONT>            manager.incrementIterationCount();<a name="line.1156"></a>
<FONT color="green">1157</FONT>    <a name="line.1157"></a>
<FONT color="green">1158</FONT>            final State state;<a name="line.1158"></a>
<FONT color="green">1159</FONT>            state = new State(a, m, b, goodb, shift, delta, check);<a name="line.1159"></a>
<FONT color="green">1160</FONT>            state.init();<a name="line.1160"></a>
<FONT color="green">1161</FONT>            state.refineSolution(x);<a name="line.1161"></a>
<FONT color="green">1162</FONT>            IterativeLinearSolverEvent event;<a name="line.1162"></a>
<FONT color="green">1163</FONT>            event = new DefaultIterativeLinearSolverEvent(this,<a name="line.1163"></a>
<FONT color="green">1164</FONT>                                                          manager.getIterations(),<a name="line.1164"></a>
<FONT color="green">1165</FONT>                                                          x,<a name="line.1165"></a>
<FONT color="green">1166</FONT>                                                          b,<a name="line.1166"></a>
<FONT color="green">1167</FONT>                                                          state.getNormOfResidual());<a name="line.1167"></a>
<FONT color="green">1168</FONT>            if (state.bEqualsNullVector()) {<a name="line.1168"></a>
<FONT color="green">1169</FONT>                /* If b = 0 exactly, stop with x = 0. */<a name="line.1169"></a>
<FONT color="green">1170</FONT>                manager.fireTerminationEvent(event);<a name="line.1170"></a>
<FONT color="green">1171</FONT>                return x;<a name="line.1171"></a>
<FONT color="green">1172</FONT>            }<a name="line.1172"></a>
<FONT color="green">1173</FONT>            /* Cause termination if beta is essentially zero. */<a name="line.1173"></a>
<FONT color="green">1174</FONT>            final boolean earlyStop;<a name="line.1174"></a>
<FONT color="green">1175</FONT>            earlyStop = state.betaEqualsZero() || state.hasConverged();<a name="line.1175"></a>
<FONT color="green">1176</FONT>            manager.fireInitializationEvent(event);<a name="line.1176"></a>
<FONT color="green">1177</FONT>            if (!earlyStop) {<a name="line.1177"></a>
<FONT color="green">1178</FONT>                do {<a name="line.1178"></a>
<FONT color="green">1179</FONT>                    manager.incrementIterationCount();<a name="line.1179"></a>
<FONT color="green">1180</FONT>                    event = new DefaultIterativeLinearSolverEvent(this,<a name="line.1180"></a>
<FONT color="green">1181</FONT>                                                                  manager.getIterations(),<a name="line.1181"></a>
<FONT color="green">1182</FONT>                                                                  x,<a name="line.1182"></a>
<FONT color="green">1183</FONT>                                                                  b,<a name="line.1183"></a>
<FONT color="green">1184</FONT>                                                                  state.getNormOfResidual());<a name="line.1184"></a>
<FONT color="green">1185</FONT>                    manager.fireIterationStartedEvent(event);<a name="line.1185"></a>
<FONT color="green">1186</FONT>                    state.update();<a name="line.1186"></a>
<FONT color="green">1187</FONT>                    state.refineSolution(x);<a name="line.1187"></a>
<FONT color="green">1188</FONT>                    event = new DefaultIterativeLinearSolverEvent(this,<a name="line.1188"></a>
<FONT color="green">1189</FONT>                                                                  manager.getIterations(),<a name="line.1189"></a>
<FONT color="green">1190</FONT>                                                                  x,<a name="line.1190"></a>
<FONT color="green">1191</FONT>                                                                  b,<a name="line.1191"></a>
<FONT color="green">1192</FONT>                                                                  state.getNormOfResidual());<a name="line.1192"></a>
<FONT color="green">1193</FONT>                    manager.fireIterationPerformedEvent(event);<a name="line.1193"></a>
<FONT color="green">1194</FONT>                } while (!state.hasConverged());<a name="line.1194"></a>
<FONT color="green">1195</FONT>            }<a name="line.1195"></a>
<FONT color="green">1196</FONT>            event = new DefaultIterativeLinearSolverEvent(this,<a name="line.1196"></a>
<FONT color="green">1197</FONT>                                                          manager.getIterations(),<a name="line.1197"></a>
<FONT color="green">1198</FONT>                                                          x,<a name="line.1198"></a>
<FONT color="green">1199</FONT>                                                          b,<a name="line.1199"></a>
<FONT color="green">1200</FONT>                                                          state.getNormOfResidual());<a name="line.1200"></a>
<FONT color="green">1201</FONT>            manager.fireTerminationEvent(event);<a name="line.1201"></a>
<FONT color="green">1202</FONT>            return x;<a name="line.1202"></a>
<FONT color="green">1203</FONT>        }<a name="line.1203"></a>
<FONT color="green">1204</FONT>    <a name="line.1204"></a>
<FONT color="green">1205</FONT>        /**<a name="line.1205"></a>
<FONT color="green">1206</FONT>         * {@inheritDoc}<a name="line.1206"></a>
<FONT color="green">1207</FONT>         *<a name="line.1207"></a>
<FONT color="green">1208</FONT>         * @param x the vector to be updated with the solution; {@code x} should<a name="line.1208"></a>
<FONT color="green">1209</FONT>         * not be considered as an initial guess (&lt;a href="#initguess"&gt;more&lt;/a&gt;)<a name="line.1209"></a>
<FONT color="green">1210</FONT>         * @throws NonSelfAdjointOperatorException if {@link #getCheck()} is<a name="line.1210"></a>
<FONT color="green">1211</FONT>         * {@code true}, and {@code a} is not self-adjoint<a name="line.1211"></a>
<FONT color="green">1212</FONT>         * @throws IllConditionedOperatorException if {@code a} is ill-conditioned<a name="line.1212"></a>
<FONT color="green">1213</FONT>         */<a name="line.1213"></a>
<FONT color="green">1214</FONT>        @Override<a name="line.1214"></a>
<FONT color="green">1215</FONT>        public RealVector solveInPlace(final RealLinearOperator a,<a name="line.1215"></a>
<FONT color="green">1216</FONT>            final RealVector b, final RealVector x) throws NullArgumentException,<a name="line.1216"></a>
<FONT color="green">1217</FONT>            NonSquareOperatorException, DimensionMismatchException,<a name="line.1217"></a>
<FONT color="green">1218</FONT>            NonSelfAdjointOperatorException, IllConditionedOperatorException,<a name="line.1218"></a>
<FONT color="green">1219</FONT>            MaxCountExceededException {<a name="line.1219"></a>
<FONT color="green">1220</FONT>            return solveInPlace(a, null, b, x, false, 0.);<a name="line.1220"></a>
<FONT color="green">1221</FONT>        }<a name="line.1221"></a>
<FONT color="green">1222</FONT>    }<a name="line.1222"></a>




























































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